{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Random numbers"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Uniform and normal distributions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To generate a random number drawn from the standard uniform distribution, we can use command `rand` from package `Random`:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.11564321538461608"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"using Random \n",
"\n",
"rand()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To get many random numbers presented in the array form, we must give the dimensions:"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×4 Matrix{Float64}:\n",
" 0.340689 0.636099 0.10403 0.763712\n",
" 0.260876 0.554632 0.184687 0.637002\n",
" 0.752655 0.0436827 0.318437 0.598123"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rand(3, 4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The same command can be used to draw from the _discrete_ uniform distribution.\n",
"To this end, as the first argument we provide the _list of elements_ as an array or a range.\n",
"All the elements are drawn with equal probabilities:"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"rand([0.2 0.3 0.5]) = 0.2\n",
"rand(100:130) = 127\n"
]
}
],
"source": [
"@show rand([.2 .3 .5]);\n",
"@show rand(100:130);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An array of the size $S\\times K\\times N$ with elements drawn from `list_of_elements` can be generated by using `rand(list_of_elements, dim1, dim2,...)`, _e.g._"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×3×2 Array{String, 3}:\n",
"[:, :, 1] =\n",
" \"b\" \"a\" \"b\"\n",
" \"c\" \"a\" \"b\"\n",
" \"b\" \"c\" \"a\"\n",
" \"b\" \"c\" \"a\"\n",
"\n",
"[:, :, 2] =\n",
" \"c\" \"a\" \"a\"\n",
" \"c\" \"b\" \"b\"\n",
" \"b\" \"c\" \"a\"\n",
" \"b\" \"b\" \"b\""
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rand([\"a\" \"b\" \"c\"], 4,3,2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Analogously, you can draw numbers from the standard normal distribution $\\mathcal{N}(\\mu=0,\\sigma^2=1)$ by using `randn`:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×3 Matrix{Float64}:\n",
" 0.55025 1.17845 -0.594802\n",
" 0.857231 0.591131 1.05621"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"randn(2,3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Multivariate normal distribution"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(multi-normal)=\n",
"```{prf:example}\n",
"Draw sample of $10000$ observations from distribution given by the multivariate nomal distribution:\n",
"\n",
"$$\\left(\\substack{y_1\\\\y_2}\\right)\\sim \\mathcal{N}\n",
"\t\\left( \\left(\\substack{2\\\\1}\\right), \\left[\n",
"\\begin{array}{cc}\n",
".75 & .1 \\\\\n",
" .1 & .25 \\\\\n",
" \\end{array}\n",
"\\right]\\right).$$\n",
"\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We know that if $x\\sim \\mathcal{N}(\\mu, \\Sigma)$ and $y=\\alpha+Bx,$ then $y\\sim\\mathcal{N}(\\alpha+B \\mu, B \\Sigma B').$ Given $x\\sim \\mathcal{N}(0, \\mathbf{I}),$ we can produce $y$ by transformation $y=\\left(\\substack{2\\\\1}\\right)+Bx,$ where $B$ is a lower triangular matrix of Cholesky decomposition for\n",
"\n",
"$$B\\Sigma B'=\\left[\n",
"\\begin{array}{cc}\n",
".75 & .1 \\\\\n",
" .1 & .25 \\\\\n",
" \\end{array}\n",
"\\right].$$\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using this property, to get $\\mathbf{y}$ we can draw two random variables $x_1$ and $x_2$ independetly from the standard normal distribution first."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10000×2 Matrix{Float64}:\n",
" 1.28189 1.38612\n",
" -1.52057 1.87422\n",
" 0.917451 0.0818603\n",
" 1.79977 2.12049\n",
" -0.395676 0.429556\n",
" -2.03951 -0.235728\n",
" 0.875222 0.059025\n",
" 1.53602 -0.2002\n",
" 1.06178 -0.575869\n",
" 1.14847 -0.387976\n",
" -0.0378603 -0.729491\n",
" 0.769667 0.0636707\n",
" -1.69662 0.375191\n",
" ⋮ \n",
" 0.255923 0.761956\n",
" -0.668536 -1.97555\n",
" -2.01672 -2.50246\n",
" 0.282025 1.74887\n",
" 0.107665 -0.107259\n",
" 1.7627 0.385254\n",
" -0.140983 -0.15348\n",
" -0.876612 -0.5456\n",
" 0.0683084 -0.138102\n",
" 1.6804 -0.166924\n",
" 0.118446 2.12958\n",
" 1.15648 1.12023"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"X = randn(10000, 2); #2 columns; each column is a r.v."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we can see the mean and variance of both variables are equal to around $0$ and $1,$ respectively.\n",
"Both variables are independent, so their covariance are close to $0.$"
]
},
{
"cell_type": "code",
"execution_count": 155,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"mean(x₁)=-0.007842019051014344\n",
"var(x₁)=1.0211957808849637\n",
"mean(x₂)=0.015231055475408468\n",
"var(x₂)=1.015896872667751\n",
"cov(x₁, x₂)=0.007499565356589907\n"
]
}
],
"source": [
"using Statistics\n",
"\n",
"println(\"mean(x₁)=$(mean(X[:,1]))\")\n",
"println(\"var(x₁)=$(var(X[:,1]))\")\n",
"\n",
"println(\"mean(x₂)=$(mean(X[:,2]))\")\n",
"println(\"var(x₂)=$(var(X[:,2]))\")\n",
"\n",
"println(\"cov(x₁, x₂)=$(cov(X[:,1], X[:,2]))\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In our example, the covariance matrix of $\\mathbf{y}$ is given by:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 LowerTriangular{Float64, Matrix{Float64}}:\n",
" 0.866025 ⋅ \n",
" 0.11547 0.486484"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"BΣBt = [.75 .1;\n",
" .1 .25]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To conduct Cholesky decomposition we use command `cholesky` from pakcage `LinearAlgebra`.\n",
"Without getting too much into details, it creates a structure with elements that can be accessed by using `.` after a created object.\n",
"To get a lower triangular matrix from the decomposition we need to call `cholesky(BBt).L`.\n",
"This function is written in the object-oriented paradigm."
]
},
{
"cell_type": "code",
"execution_count": 156,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 LowerTriangular{Float64, Matrix{Float64}}:\n",
" 0.866025 ⋅ \n",
" 0.11547 0.486484"
]
},
"execution_count": 156,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using LinearAlgebra\n",
"\n",
"B = cholesky(BΣBt).L"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Indeed, our new matrix, $B$, is the matrix we are looking for as _i.e._\n",
"\n",
"$$B B'=\\left[\n",
"\\begin{array}{cc}\n",
".75 & .1 \\\\\n",
" .1 & .25 \\\\\n",
" \\end{array}\n",
"\\right].$$"
]
},
{
"cell_type": "code",
"execution_count": 157,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Matrix{Float64}:\n",
" 0.75 0.1\n",
" 0.1 0.25"
]
},
"execution_count": 157,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B*B'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To get $\\mathbf{y},$ for each pair $(x_1, x_2)$ we need to perform the following operation $\\left(\\substack{y_1\\\\y_2}\\right) =\\left(\\substack{2\\\\1}\\right)+B\\left(\\substack{x_1\\\\x_2}\\right).$\n",
"We can use `map` for this:"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10000-element Vector{Matrix{Float64}}:\n",
" [2.1101528096781395 2.8223440507646176]\n",
" [-0.316852269162089 2.7361982939723863]\n",
" [1.7945360086064883 2.145761870909989]\n",
" [2.5586476794344764 3.239405975767194]\n",
" [0.6573344169912505 2.163283500765393]\n",
" [-0.7662691970149982 1.649819693668626]\n",
" [1.7579640573520672 2.1297765784825953]\n",
" [2.3302337351792275 2.0799703605388933]\n",
" [1.91953104210994 1.8424529213877525]\n",
" [1.9946041051183943 1.9438696567103626]\n",
" [0.9672120594130107 1.6407425578761052]\n",
" [1.6665511575809437 2.119848254067102]\n",
" [-0.4693134642601875 1.9866161385776993]\n",
" ⋮\n",
" [1.2216356982679555 2.400230785501348]\n",
" [0.4210307841448212 0.9617287397615868]\n",
" [-0.7465331243930957 0.5497224297388332]\n",
" [1.2442410317329406 2.883361465346356]\n",
" [1.093240765317267 1.9602525055505302]\n",
" [2.5265399280298535 2.390958398962555]\n",
" [0.8779052687865513 1.9090552115558233]\n",
" [0.24083211563820572 1.633352166745183]\n",
" [1.0591567947011478 1.9407029390230028]\n",
" [2.455268286364639 2.1128296859828852]\n",
" [1.1025774669099702 3.0496823940530273]\n",
" [2.001536829752592 2.6785125301903085]"
]
},
"execution_count": 160,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y_map = map( (x₁, x₂) ->begin\n",
" x_vec = [x₁, x₂]\n",
" y₁, y₂ = [1, 2] + B*x_vec\n",
" return [y₁ y₂]\n",
" end,\n",
" X[:,1], X[:,2])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The returned object is an array with two-element vectors $(y_1, y_2)$.\n",
"For convenience, we may want to extract those elements and arrange $(y_1, y_2)$ in a two-column array:"
]
},
{
"cell_type": "code",
"execution_count": 169,
"metadata": {},
"outputs": [],
"source": [
"Y = Array{Float64}(undef, 10000,2);\n",
"Y[:, 1] = map(x-> x[1] ,y_map);\n",
"Y[:, 2] = map(x-> x[2] ,y_map);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we see, means and covariances are approximately equal to values of the random variables from which we wanted to draw random numbers:"
]
},
{
"cell_type": "code",
"execution_count": 172,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"mean(y₁)=0.9932086122848602\n",
"var(y₁)=0.7658968356637227\n",
"mean(y₂)=2.0065041461858413\n",
"var(y₂)=0.254887436422212\n",
"cov(y₁, y₂) = 0.10527920113491405\n"
]
}
],
"source": [
"println(\"mean(y₁)=$(mean(Y[:,1]))\")\n",
"println(\"var(y₁)=$(var(Y[:,1]))\")\n",
"\n",
"println(\"mean(y₂)=$(mean(Y[:,2]))\")\n",
"println(\"var(y₂)=$(var(Y[:,2]))\")\n",
"\n",
"println(\"cov(y₁, y₂) = $(cov(Y[:,1], Y[:,2]))\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Instead of using `map`, we could perform simple matrix operations.\n",
"The result would be exactly the same:"
]
},
{
"cell_type": "code",
"execution_count": 173,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10000×2 Matrix{Float64}:\n",
" 2.11015 2.82234\n",
" -0.316852 2.7362\n",
" 1.79454 2.14576\n",
" 2.55865 3.23941\n",
" 0.657334 2.16328\n",
" -0.766269 1.64982\n",
" 1.75796 2.12978\n",
" 2.33023 2.07997\n",
" 1.91953 1.84245\n",
" 1.9946 1.94387\n",
" 0.967212 1.64074\n",
" 1.66655 2.11985\n",
" -0.469313 1.98662\n",
" ⋮ \n",
" 1.22164 2.40023\n",
" 0.421031 0.961729\n",
" -0.746533 0.549722\n",
" 1.24424 2.88336\n",
" 1.09324 1.96025\n",
" 2.52654 2.39096\n",
" 0.877905 1.90906\n",
" 0.240832 1.63335\n",
" 1.05916 1.9407\n",
" 2.45527 2.11283\n",
" 1.10258 3.04968\n",
" 2.00154 2.67851"
]
},
"execution_count": 173,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1 2] .+ X*B'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Simulating Markov chains"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(markov-simulation)=\n",
"```{prf:example}\n",
"Suppose that the income of a single household follows a certain Markov chain with the transition probability: \n",
"\n",
"$$\\mathbb{P}=\\begin{bmatrix}\n",
".8 & .1 & .1\\\\\n",
".1 & .8 & .1\\\\\n",
".1 & .1 & .8\\\\\n",
"\\end{bmatrix}$$ \n",
"\n",
"and income states equal to $\\mathbf{y} = [e^{-\\frac{1}{2}}, 1 , e^{\\frac{1}{2}}].$\n",
"\n",
"Simulate the dynamics of the income in the horizon of $100$ periods. The initial income is equal to $y_0=1.$\n",
"\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Command `rand` allows sampling from a finite set of elements where element has equal probabilities.\n",
"A similar thing can be done using `sample` from package `StatsBase`:"
]
},
{
"cell_type": "code",
"execution_count": 179,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.6065306597126334"
]
},
"execution_count": 179,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sample( exp.([-.5, 0, .5]) )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As in the previous example, the probability is the same across elements:\n",
"\n",
"```{margin}\n",
"`countmap` computes the frequency of each element of an array.\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 188,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Float64, Int64} with 3 entries:\n",
" 1.64872 => 3356\n",
" 1.0 => 3380\n",
" 0.606531 => 3264"
]
},
"execution_count": 188,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"draws = [sample(exp.([-.5, 0, .5])) for i ∈1:10000] # sample(exp.([-.5, 0, .5]) , 10000 ) will give the same result\n",
"countmap( draws )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we want to draw element with different probabilities, we have provide weights:\n"
]
},
{
"cell_type": "code",
"execution_count": 191,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Float64, Int64} with 3 entries:\n",
" 1.64872 => 956\n",
" 1.0 => 968\n",
" 0.606531 => 8076"
]
},
"execution_count": 191,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"draws = [sample( exp.([-.5, 0, .5]), Weights( [.8, .1, .1] ) ) for i ∈1:10000]\n",
"countmap( draws )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the considered example, the probabilities of states in the next period depends on the current state.\n",
"The transition matrix is given by:"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Matrix{Float64}:\n",
" 0.8 0.1 0.1\n",
" 0.1 0.8 0.1\n",
" 0.1 0.1 0.8"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P = [.8 .1 .1; \n",
" .1 .8 .1; \n",
" .1 .1 .8]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To simulate the income dynamics we need to build a loop in which `Weights` are taken from a row of the transition matrix associated with the current state:"
]
},
{
"cell_type": "code",
"execution_count": 227,
"metadata": {},
"outputs": [],
"source": [
"history = Array{Int}(undef, 101); #index (!) of the current state\n",
"history[1] = 2; \n",
"\n",
"using StatsBase\n",
"\n",
"\n",
"for τ ∈ 2:length(history)\n",
" history[τ] = sample(1:length(y), \n",
" Weights( P[history[τ-1], :] )\n",
" )\n",
"\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Eventually, we have one simulated dynamics of the Markov chain:"
]
},
{
"cell_type": "code",
"execution_count": 193,
"metadata": {},
"outputs": [
{
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},
"execution_count": 193,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"plot(y[history])\n",
"ylabel!(\"Income\")\n",
"xlabel!(\"Time\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Random numbers generated through a quantile function"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Continuous cdfs"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(Fx_x2)=\n",
"```{prf:example}\n",
"\n",
"Suppose that we want to draw random numbers according to a distribution given by the following cumulative distribution function:\n",
"\n",
"\n",
"$$ F(x) = \\begin{cases} \n",
" 0, \\text{if $x< 0$ }\\\\\n",
" x^2, \\text{if $x\\in[0,1]$ }\\\\\n",
" 1, \\text{if $x>1$ }\n",
" \\end{cases}$$\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function has different definitions for three intervals, $(-\\infty, 0), [0,1], (1,+\\infty)$.\n",
"Using a similar approach to [the one](crra) used in creating the CRRA utility function, we can use ternary operator `?:` twice to create $F(x)$ in `Julia`:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"F (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"F(x) = 0≤x≤1 ? x^2 :\n",
" ( x< 0 ? 0 : 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we can see, $F(x)$ continuous and stricly increasing for $x\\in[0,1].$ \n",
"````{margin}\n",
"Notice that here we provide a _function_ as the input. Alternatively, we could build our own range, which would be broadcasted to $F(x)$: \n",
"\n",
"```\n",
"using Plots\n",
"using Pipe\n",
"@pipe range(-.1, 1.2, \n",
" length=100) |> \n",
" plot(_, F.(_))\n",
"```\n",
"````"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"plot(F, -.1, 1.2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that the values $F(x)$ of each continuous cdf is uniformly distributed.\n",
"This means that each value $p\\in[0,1]$ is equally likely.\n",
"To draw random numbers for a random variable distributed according to $F(x),$ we can use the quantile function (also called the inverse cdf).\n",
"This function maps quantiles to a threshold value $x$ below which random draws would fall $p$ percent of the time.\n",
"Next, draws from the uniform distribution can be used to represent quantiles of the distribution given by $F(x).$\n",
"Using the quantile function, those draws can be projected on actual values, $x.$\n",
" \n",
"For monotonically increasing and continuous cdf's, this is simply the inverse function of the cdf.\n",
"In our case, it is:\n",
"\n",
"$$\n",
"Q(p) = \\sqrt{p},\n",
"$$\n",
"\n",
"where $p\\in[0,1].$ \n",
"In `Julia` we can define this function in the following way:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"invF (generic function with 1 method)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"invF(p) = 0≤p≤1 ? √(p) : @error \"Forbidden argument: p∉[0,1]\" "
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
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},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(invF, 0, 1)\n",
"xlabel!(\"p\"); ylabel!(\"x\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can generate some numbers from the uniform distribution and then by using `invF` we project those draws on the actual values of $x$. We do this exercise for $10,000$ draws and $20$ draws:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
"draws = rand(10000);\n",
"draws_x = invF.(draws);"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [],
"source": [
"draws_very_few = rand(20);\n",
"draws_x_very_few = invF.(draws_very_few);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, let's plot the empirical cdf's of numbers drawn from the uniform distribution and compare them with the true cdf (which for the standardized uniform distribution is equal to $G(x)=x$).\n",
"To get empirical cdf's, I will use `ecdf` from package `StatsBase`.\n",
"The output of `ecdf` is a _function_ that returns the implied values of the cdf for _all_ values (not only the drawn ones).\n"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [],
"source": [
"using StatsBase\n",
"draws_ecdf = ecdf(draws);\n",
"draws_very_few_ecdf = ecdf(draws_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.4986"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"draws_ecdf(.5)"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×3 Matrix{Float64}:\n",
" 0.3 0.6 0.75"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"draws_very_few_ecdf.( [.2 .4 .6] )"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
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},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"using Pipe\n",
"@pipe range(0,1, length=100) |> plot(_, draws_ecdf.(_), label=\"Empirical CDF (N=10,000)\",\n",
" legend = :topleft)\n",
"@pipe range(0,1, length=100) |> plot!(_, draws_very_few_ecdf.(_), label =\"Empirical CDF (N=20)\")\n",
"@pipe range(0,1, length=100) |> plot!(_, map( x-> x, _), label =\"True CDF of the uniform dist.\")\n",
"title!(\"Draws of quantiles vs. cdf of the uniform dist.\")\n",
"xlabel!(\"p\")\n",
"ylabel!(\"cdf\")\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As can be seen, the empirical cdf with 10,000 draws resembles the true cdf of the uniform distribution very closely, while there is a much bigger discrepancy for the simulation only with 20 observations.\n",
"\n",
"Now let's compare the empirical cdf of implied $x$ for 10,000 and 20 draws with the true cdf, the quadratic function from [our example](Fx_x2)."
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [],
"source": [
"using StatsBase\n",
"cdf_x = ecdf(draws_x);\n",
"cdf_x_very_few = ecdf(draws_x_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
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},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Pipe\n",
"@pipe range(0,1, length=100) |> plot(_, cdf_x.(_), label=\"Empirical CDF (N=10000)\",\n",
" legend=:topleft)\n",
"@pipe range(0,1, length=100) |> plot!(_, cdf_x_very_few.(_), label =\"Empirical CDF (N=20)\")\n",
"@pipe range(0,1, length=100) |> plot!(_, F.(_), label =\"True CDF\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Our exercise shows that by using the quantile function and random number generators from the uniform distribution, we can draw numbers from any arbitrary distribution, which exhibits continuous cdf and its quantile function is closed-form. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Continuous cdf but no closed-formed solution for the quantile function\n",
"\n",
"(Gx_cdf)=\n",
"```{prf:example}\n",
"\n",
"Suppose that we want to draw random numbers according to a distribution given by the following cumulative distribution function:\n",
"\n",
"\n",
"$$ G(x) = \\begin{cases} \n",
" 0, \\text{if $x< 0$ }\\\\\n",
" \\frac{x^2+x\\sqrt{x+3}}{3}, \\text{if $x\\in[0,1]$ }\\\\\n",
" 1, \\text{if $x>1$ }\n",
" \\end{cases}$$\n",
"```\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Function $G(x)$ is a continuous cdf just like in our previous example:"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
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},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G(x) = 0≤x≤1 ? ( x^2 + x * √(x+3) )/3 :\n",
" ( x< 0 ? 0 : 1)\n",
"\n",
"plot(G, -.1, 1.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similarly to $F(x)$, $G(x)$ is a continuous cdf. \n",
"In contrast to our previous example, it is not easy to derive the inverse function of $G(x)$ in the closed form.\n",
"Recall, for a given quantile $p^*,$ we want to find such $x$ that meets $p^* = F(x)$.\n",
"Therefore, the quantile function is defined as:\n",
"\n",
"$$\n",
"Q(p^*) = \\{x\\in \\mathbb{R}:\\quad p^* = F(x) \\}.\n",
"$$\n",
"\n",
"This issue can be addressed numerically.\n",
"We can run a root finding procedure for the function given by: \n",
"\n",
"$$f(x) = p^* - F(x),$$\n",
"\n",
"where $p^*$ is the parameter.\n",
"\n",
"Let's ilustrate this by find the median of the distribution given by $G(x).$\n",
"For this, we can use function `find_zero` from package `Roots`:"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.6004852578317297"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Roots\n",
"median = find_zero( x -> .5 - G(x), .1) #2nd argument: initial guess for $x$"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.6004852578317297"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"median"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For the distribution given by $G(x)$, the median is equal to around $0.6.$\n",
"We can illustrate this by using a plot:"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
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},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(G, -.1, 1.1, label=\"cdf\", \n",
" legend=:topleft)\n",
"plot!([-.1, median], [.5, .5], label=\"value of p*\")\n",
"plot!([median, median], [0, .5], label=\"root of the cdf given p*\")\n",
"xlabel!(\"x\")\n",
"ylabel!(\"p\")\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"```{note}\n",
"More generally, we can use function `invG(p)`, which for a given input `p` conducts root finding procedure `find_zero( x -> p - G(x), .1)`.\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"invG (generic function with 1 method)"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"invG(p) = find_zero( x -> p - G(x), .1)"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [],
"source": [
"draws = rand(10000);\n",
"draws_G = invG.(draws);\n",
"\n",
"draws_very_few = rand(20);\n",
"draws_G_very_few = invG.(draws_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [],
"source": [
"ecdf_G = ecdf(draws_G);\n",
"ecdf_G_very_few = ecdf(draws_G_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {},
"outputs": [
{
"data": {
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},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@pipe range(0,1, length=100) |> plot(_, ecdf_G.(_), label=\"Empirical CDF (N=10000)\",\n",
" legend=:topleft)\n",
"@pipe range(0,1, length=100) |> plot!(_, ecdf_G_very_few.(_), label =\"Empirical CDF (N=20)\")\n",
"@pipe range(0,1, length=100) |> plot!(_, G.(_), label =\"True CDF\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Cdfs with discrete jumps"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(Hx_discrete)=\n",
"```{prf:example}\n",
"\n",
"Suppose that we want to draw random numbers according to a distribution given by the following cumulative distribution function:\n",
"\n",
"\n",
"$$ H(x) = \\begin{cases} \n",
" 0, \\text{if $x< 0$ }\\\\\n",
" x^3, \\text{if $x\\in [0,\\frac{1}{2})$ }\\\\\n",
" x^2, \\text{if $x\\in[\\frac{1}{2},1]$}\\\\\n",
" 1, \\text{if $x>1$ }\n",
" \\end{cases}\n",
" $$\n",
" \n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Just like before, we can define the cdf using `?.` ternary operator:"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"H (generic function with 1 method)"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"H(x) = 0≤x≤1 ? ( x <.5 ? x^3 : x^2) :\n",
" ( x < 0 ? 0 : 1)"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {},
"outputs": [
{
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},
"execution_count": 117,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(H, -.1, 1.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"````{margin}\n",
"\n",
"`Plots` is not aware of the discontinuity at $0.5.$\n",
"To plot correctly, we need to draw fragments of the function separately:\n",
"\n",
"```\n",
"plot(H, -.1, .49, legend=false, color=\"blue\")\n",
"plot!(H, .5, 1.1, color=\"blue\")\n",
"```\n",
"That being said, in further analysis I abstract from this violation of the defined function in plots.\n",
"\n",
"\n",
"\n",
"````\n",
"\n",
"In the considered function, there is a discrete jump at $x=0.5,$ which means that there is a probability mass at this point. \n",
"In our case, there is the likelihood of $\\left({.5}\\right)^2-\\left({.5}\\right)^3=.125$ that the drawn number will be equal exactly to $.5.$\n",
"\n",
"For distributions with discrete jumps, we need to use a generalized version of the quantile function:\n",
"\n",
"$$ \n",
"Q(p)\\,=\\,\\inf\\left\\{ x\\in \\mathbb{R} : p \\le F(x) \\right\\}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To study distributions with discrete jumps\n",
"\n",
"$$ \n",
"Q(p)\\,=\\,\\inf\\left\\{ x\\in \\mathbb{R} : p \\le F(x) \\right\\}\n",
"$$\n",
"\n",
"Consequently, the (extended) quantile for the one from the [discussed example](Hx_discrete) will be as follows: \n",
"\n",
"$$ Q(p) = \\begin{cases} \n",
" \n",
" \\sqrt[3]{p}, \\text{if $p\\in [0,\\frac{1}{8})$ }\\\\\n",
" 0.5, \\text{if $p\\in [\\frac{1}{8}, \\frac{1}{4})$ }\\\\\n",
" \\sqrt{p}, \\text{if $p\\in(\\frac{1}{4},1]$}\\\\\n",
" \n",
" \\end{cases}\n",
" $$\n"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"invH (generic function with 1 method)"
]
},
"execution_count": 139,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"invH(p) = 0≤p≤1 ? ( p<.125 ? ∛(p) : \n",
" .125 ≤ p ≤ .25 ? .5 :\n",
" √(p) ) : \n",
" @error \"Forbidden argument: p∉[0,1]\" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"All other steps will be exactly the same as in [the initial example](Fx_x2) with a continuous cdf."
]
},
{
"cell_type": "code",
"execution_count": 140,
"metadata": {},
"outputs": [],
"source": [
"draws = rand(10000);\n",
"draws_H = invH.(draws);\n",
"\n",
"draws_very_few = rand(20);\n",
"draws_H_very_few = invH.(draws_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [],
"source": [
"ecdf_H = ecdf(draws_H);\n",
"ecdf_H_very_few = ecdf(draws_H_very_few);"
]
},
{
"cell_type": "code",
"execution_count": 144,
"metadata": {},
"outputs": [
{
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497.189,1443.24 499.235,1443.18 501.281,1443.12 503.327,1443.06 505.373,1442.99 507.418,1442.93 509.464,1442.86 511.51,1442.79 \n 513.556,1442.72 515.602,1442.65 517.648,1442.58 519.694,1442.51 521.74,1442.44 523.786,1442.37 525.831,1442.29 527.877,1442.21 529.923,1442.14 531.969,1442.06 \n 534.015,1441.98 536.061,1441.9 538.107,1441.82 540.153,1441.73 542.199,1441.65 544.245,1441.56 546.29,1441.48 548.336,1441.39 550.382,1441.3 552.428,1441.21 \n 554.474,1441.12 556.52,1441.03 558.566,1440.93 560.612,1440.84 562.658,1440.74 564.704,1440.64 566.749,1440.55 568.795,1440.45 570.841,1440.34 572.887,1440.24 \n 574.933,1440.14 576.979,1440.03 579.025,1439.93 581.071,1439.82 583.117,1439.71 585.162,1439.6 587.208,1439.49 589.254,1439.37 591.3,1439.26 593.346,1439.14 \n 595.392,1439.03 597.438,1438.91 599.484,1438.79 601.53,1438.66 603.576,1438.54 605.621,1438.42 607.667,1438.29 609.713,1438.16 611.759,1438.04 613.805,1437.91 \n 615.851,1437.77 617.897,1437.64 619.943,1437.51 621.989,1437.37 624.035,1437.23 626.08,1437.09 628.126,1436.95 630.172,1436.81 632.218,1436.67 634.264,1436.52 \n 636.31,1436.37 638.356,1436.23 640.402,1436.08 642.448,1435.93 644.493,1435.77 646.539,1435.62 648.585,1435.46 650.631,1435.3 652.677,1435.14 654.723,1434.98 \n 656.769,1434.82 658.815,1434.66 660.861,1434.49 662.907,1434.32 664.952,1434.15 666.998,1433.98 669.044,1433.81 671.09,1433.64 673.136,1433.46 675.182,1433.28 \n 677.228,1433.1 679.274,1432.92 681.32,1432.74 683.366,1432.56 685.411,1432.37 687.457,1432.18 689.503,1431.99 691.549,1431.8 693.595,1431.61 695.641,1431.41 \n 697.687,1431.22 699.733,1431.02 701.779,1430.82 703.825,1430.61 705.87,1430.41 707.916,1430.2 709.962,1430 712.008,1429.79 714.054,1429.57 716.1,1429.36 \n 718.146,1429.15 720.192,1428.93 722.238,1428.71 724.283,1428.49 726.329,1428.27 728.375,1428.04 730.421,1427.82 732.467,1427.59 734.513,1427.36 736.559,1427.12 \n 738.605,1426.89 740.651,1426.65 742.697,1426.42 744.742,1426.18 746.788,1425.93 748.834,1425.69 750.88,1425.44 752.926,1425.19 754.972,1424.94 757.018,1424.69 \n 759.064,1424.44 761.11,1424.18 763.156,1423.92 765.201,1423.66 767.247,1423.4 769.293,1423.13 771.339,1422.87 773.385,1422.6 775.431,1422.33 777.477,1422.06 \n 779.523,1421.78 781.569,1421.5 783.614,1421.22 785.66,1420.94 787.706,1420.66 789.752,1420.37 791.798,1420.08 793.844,1419.79 795.89,1419.5 797.936,1419.21 \n 799.982,1418.91 802.028,1418.61 804.073,1418.31 806.119,1418.01 808.165,1417.7 810.211,1417.39 812.257,1417.08 814.303,1416.77 816.349,1416.46 818.395,1416.14 \n 820.441,1415.82 822.487,1415.5 824.532,1415.18 826.578,1414.85 828.624,1414.52 830.67,1414.19 832.716,1413.86 834.762,1413.52 836.808,1413.18 838.854,1412.84 \n 840.9,1412.5 842.945,1412.16 844.991,1411.81 847.037,1411.46 849.083,1411.11 851.129,1410.75 853.175,1410.4 855.221,1410.04 857.267,1409.68 859.313,1409.31 \n 861.359,1408.95 863.404,1408.58 865.45,1408.21 867.496,1407.83 869.542,1407.46 871.588,1407.08 873.634,1406.7 875.68,1406.31 877.726,1405.93 879.772,1405.54 \n 881.818,1405.15 883.863,1404.75 885.909,1404.36 887.955,1403.96 890.001,1403.55 892.047,1403.15 894.093,1402.74 896.139,1402.33 898.185,1401.92 900.231,1401.51 \n 902.276,1401.09 904.322,1400.67 906.368,1400.25 908.414,1399.82 910.46,1399.4 912.506,1398.97 914.552,1398.53 916.598,1398.1 918.644,1397.66 920.69,1397.22 \n 922.735,1396.78 924.781,1396.33 926.827,1395.88 928.873,1395.43 930.919,1394.97 932.965,1394.52 935.011,1394.06 937.057,1393.6 939.103,1393.13 941.149,1392.66 \n 943.194,1392.19 945.24,1391.72 947.286,1391.24 949.332,1390.76 951.378,1390.28 953.424,1389.79 955.47,1389.31 957.516,1388.82 959.562,1388.32 961.607,1387.83 \n 963.653,1387.33 965.699,1386.83 967.745,1386.32 969.791,1385.81 971.837,1385.3 973.883,1384.79 975.929,1384.27 977.975,1383.75 980.021,1383.23 982.066,1382.71 \n 984.112,1382.18 986.158,1381.65 988.204,1381.11 990.25,1380.58 992.296,1380.04 994.342,1379.49 996.388,1378.95 998.434,1378.4 1000.48,1377.85 1002.53,1377.29 \n 1004.57,1376.74 1006.62,1376.17 1008.66,1375.61 1010.71,1375.04 1012.75,1374.47 1014.8,1373.9 1016.85,1373.32 1018.89,1372.75 1020.94,1372.16 1022.98,1371.58 \n 1025.03,1370.99 1027.08,1370.4 1029.12,1369.8 1031.17,1369.21 1033.21,1368.61 1035.26,1368 1037.31,1367.39 1039.35,1366.78 1041.4,1366.17 1043.44,1365.55 \n 1045.49,1364.93 1047.54,1364.31 1049.58,1363.68 1051.63,1363.06 1053.67,1362.42 1055.72,1361.79 1057.76,1361.15 1059.81,1360.51 1061.86,1359.86 1063.9,1359.21 \n 1065.95,1358.56 1067.99,1357.9 1070.04,1357.25 1072.09,1356.58 1074.13,1355.92 1076.18,1355.25 1078.22,1354.58 1080.27,1353.9 1082.32,1353.22 1084.36,1352.54 \n 1086.41,1351.86 1088.45,1351.17 1090.5,1350.48 1092.54,1349.78 1094.59,1349.08 1096.64,1348.38 1098.68,1347.68 1100.73,1346.97 1102.77,1346.26 1104.82,1345.54 \n 1106.87,1344.82 1108.91,1344.1 1110.96,1343.37 1113,1342.64 1115.05,1341.91 1117.1,1341.17 1119.14,1340.43 1121.19,1339.69 1123.23,1338.95 1125.28,1338.2 \n 1127.33,1337.44 1129.37,1336.68 1131.42,1335.92 1133.46,1335.16 1135.51,1334.39 1137.55,1333.62 1139.6,1332.85 1141.65,1332.07 1143.69,1331.29 1145.74,1330.5 \n 1147.78,1329.71 1149.83,1328.92 1151.88,1328.12 1153.92,1327.32 1155.97,1326.52 1158.01,1325.71 1160.06,1324.9 1162.11,1324.09 1164.15,1323.27 1166.2,1322.45 \n 1168.24,1321.62 1170.29,1320.79 1172.33,1319.96 1174.38,1319.12 1176.43,1318.28 1178.47,1317.44 1180.52,1316.59 1182.56,1315.74 1184.61,1314.88 1186.66,1314.02 \n 1188.7,1313.16 1190.75,1312.3 1192.79,1311.43 1194.84,1310.55 1196.89,1309.67 1198.93,1308.79 1200.98,1307.91 1203.02,1307.02 1205.07,1306.12 1207.12,1305.23 \n 1209.16,1304.33 1211.21,1303.42 1213.25,1302.51 1215.3,1301.6 1217.34,1300.69 1219.39,1299.77 1221.44,1298.84 1223.48,1297.92 1225.53,1296.98 1227.57,1296.05 \n 1229.62,1295.11 1231.67,1294.17 1233.71,1293.22 1235.76,1292.27 1237.8,1291.31 1239.85,1290.35 1241.9,1289.39 1243.94,1288.42 1245.99,1287.45 1248.03,1286.48 \n 1250.08,1285.5 1252.12,1284.52 1254.17,1283.53 1256.22,1282.54 1258.26,1281.54 1260.31,1280.54 1262.35,1279.54 1264.4,1278.53 1266.45,1277.52 1268.49,1276.51 \n 1270.54,1105.6 1272.58,1104.24 1274.63,1102.87 1276.68,1101.51 1278.72,1100.14 1280.77,1098.76 1282.81,1097.39 1284.86,1096.01 1286.91,1094.63 1288.95,1093.25 \n 1291,1091.86 1293.04,1090.47 1295.09,1089.08 1297.13,1087.69 1299.18,1086.29 1301.23,1084.89 1303.27,1083.49 1305.32,1082.08 1307.36,1080.67 1309.41,1079.26 \n 1311.46,1077.85 1313.5,1076.43 1315.55,1075.01 1317.59,1073.59 1319.64,1072.17 1321.69,1070.74 1323.73,1069.31 1325.78,1067.88 1327.82,1066.44 1329.87,1065 \n 1331.91,1063.56 1333.96,1062.12 1336.01,1060.67 1338.05,1059.22 1340.1,1057.77 1342.14,1056.32 1344.19,1054.86 1346.24,1053.4 1348.28,1051.94 1350.33,1050.47 \n 1352.37,1049.01 1354.42,1047.54 1356.47,1046.06 1358.51,1044.59 1360.56,1043.11 1362.6,1041.63 1364.65,1040.14 1366.7,1038.65 1368.74,1037.16 1370.79,1035.67 \n 1372.83,1034.18 1374.88,1032.68 1376.92,1031.18 1378.97,1029.68 1381.02,1028.17 1383.06,1026.66 1385.11,1025.15 1387.15,1023.63 1389.2,1022.12 1391.25,1020.6 \n 1393.29,1019.08 1395.34,1017.55 1397.38,1016.02 1399.43,1014.49 1401.48,1012.96 1403.52,1011.42 1405.57,1009.88 1407.61,1008.34 1409.66,1006.8 1411.7,1005.25 \n 1413.75,1003.7 1415.8,1002.15 1417.84,1000.6 1419.89,999.038 1421.93,997.477 1423.98,995.914 1426.03,994.348 1428.07,992.779 1430.12,991.208 1432.16,989.634 \n 1434.21,988.057 1436.26,986.478 1438.3,984.896 1440.35,983.311 1442.39,981.723 1444.44,980.133 1446.49,978.54 1448.53,976.944 1450.58,975.345 1452.62,973.744 \n 1454.67,972.14 1456.71,970.533 1458.76,968.924 1460.81,967.312 1462.85,965.697 1464.9,964.079 1466.94,962.459 1468.99,960.836 1471.04,959.21 1473.08,957.582 \n 1475.13,955.95 1477.17,954.317 1479.22,952.68 1481.27,951.041 1483.31,949.398 1485.36,947.754 1487.4,946.106 1489.45,944.456 1491.49,942.803 1493.54,941.147 \n 1495.59,939.489 1497.63,937.828 1499.68,936.164 1501.72,934.497 1503.77,932.828 1505.82,931.156 1507.86,929.481 1509.91,927.804 1511.95,926.124 1514,924.441 \n 1516.05,922.755 1518.09,921.067 1520.14,919.376 1522.18,917.682 1524.23,915.986 1526.27,914.286 1528.32,912.584 1530.37,910.88 1532.41,909.172 1534.46,907.462 \n 1536.5,905.75 1538.55,904.034 1540.6,902.316 1542.64,900.595 1544.69,898.871 1546.73,897.145 1548.78,895.415 1550.83,893.684 1552.87,891.949 1554.92,890.212 \n 1556.96,888.472 1559.01,886.729 1561.06,884.983 1563.1,883.235 1565.15,881.484 1567.19,879.731 1569.24,877.974 1571.28,876.215 1573.33,874.454 1575.38,872.689 \n 1577.42,870.922 1579.47,869.152 1581.51,867.379 1583.56,865.604 1585.61,863.826 1587.65,862.045 1589.7,860.261 1591.74,858.475 1593.79,856.686 1595.84,854.894 \n 1597.88,853.1 1599.93,851.303 1601.97,849.503 1604.02,847.7 1606.06,845.895 1608.11,844.087 1610.16,842.276 1612.2,840.462 1614.25,838.646 1616.29,836.827 \n 1618.34,835.006 1620.39,833.181 1622.43,831.354 1624.48,829.524 1626.52,827.692 1628.57,825.857 1630.62,824.019 1632.66,822.178 1634.71,820.334 1636.75,818.488 \n 1638.8,816.639 1640.85,814.788 1642.89,812.934 1644.94,811.077 1646.98,809.217 1649.03,807.354 1651.07,805.489 1653.12,803.621 1655.17,801.751 1657.21,799.877 \n 1659.26,798.001 1661.3,796.122 1663.35,794.241 1665.4,792.357 1667.44,790.47 1669.49,788.58 1671.53,786.688 1673.58,784.792 1675.63,782.895 1677.67,780.994 \n 1679.72,779.091 1681.76,777.185 1683.81,775.276 1685.85,773.365 1687.9,771.45 1689.95,769.534 1691.99,767.614 1694.04,765.692 1696.08,763.767 1698.13,761.839 \n 1700.18,759.908 1702.22,757.975 1704.27,756.039 1706.31,754.101 1708.36,752.159 1710.41,750.215 1712.45,748.268 1714.5,746.319 1716.54,744.366 1718.59,742.411 \n 1720.64,740.454 1722.68,738.493 1724.73,736.53 1726.77,734.564 1728.82,732.596 1730.86,730.624 1732.91,728.65 1734.96,726.674 1737,724.694 1739.05,722.712 \n 1741.09,720.727 1743.14,718.739 1745.19,716.749 1747.23,714.756 1749.28,712.76 1751.32,710.762 1753.37,708.761 1755.42,706.757 1757.46,704.75 1759.51,702.74 \n 1761.55,700.728 1763.6,698.714 1765.64,696.696 1767.69,694.676 1769.74,692.653 1771.78,690.627 1773.83,688.599 1775.87,686.567 1777.92,684.533 1779.97,682.497 \n 1782.01,680.458 1784.06,678.415 1786.1,676.371 1788.15,674.323 1790.2,672.273 1792.24,670.22 1794.29,668.164 1796.33,666.106 1798.38,664.045 1800.43,661.981 \n 1802.47,659.915 1804.52,657.845 1806.56,655.773 1808.61,653.699 1810.65,651.621 1812.7,649.541 1814.75,647.458 1816.79,645.373 1818.84,643.284 1820.88,641.193 \n 1822.93,639.1 1824.98,637.003 1827.02,634.904 1829.07,632.802 1831.11,630.697 1833.16,628.59 1835.21,626.48 1837.25,624.367 1839.3,622.252 1841.34,620.133 \n 1843.39,618.012 1845.43,615.889 1847.48,613.762 1849.53,611.633 1851.57,609.501 1853.62,607.367 1855.66,605.23 1857.71,603.09 1859.76,600.947 1861.8,598.801 \n 1863.85,596.653 1865.89,594.502 1867.94,592.349 1869.99,590.192 1872.03,588.033 1874.08,585.872 1876.12,583.707 1878.17,581.54 1880.22,579.37 1882.26,577.197 \n 1884.31,575.022 1886.35,572.844 1888.4,570.663 1890.44,568.48 1892.49,566.293 1894.54,564.104 1896.58,561.913 1898.63,559.718 1900.67,557.521 1902.72,555.321 \n 1904.77,553.119 1906.81,550.913 1908.86,548.705 1910.9,546.494 1912.95,544.281 1915,542.065 1917.04,539.846 1919.09,537.624 1921.13,535.4 1923.18,533.173 \n 1925.22,530.943 1927.27,528.711 1929.32,526.475 1931.36,524.237 1933.41,521.997 1935.45,519.753 1937.5,517.507 1939.55,515.258 1941.59,513.007 1943.64,510.753 \n 1945.68,508.496 1947.73,506.236 1949.78,503.973 1951.82,501.708 1953.87,499.44 1955.91,497.17 1957.96,494.896 1960.01,492.62 1962.05,490.342 1964.1,488.06 \n 1966.14,485.776 1968.19,483.489 1970.23,481.199 1972.28,478.907 1974.33,476.612 1976.37,474.314 1978.42,472.013 1980.46,469.71 1982.51,467.404 1984.56,465.096 \n 1986.6,462.784 1988.65,460.47 1990.69,458.153 1992.74,455.833 1994.79,453.511 1996.83,451.186 1998.88,448.858 2000.92,446.528 2002.97,444.195 2005.01,441.859 \n 2007.06,439.52 2009.11,437.179 2011.15,434.835 2013.2,432.488 2015.24,430.139 2017.29,427.786 2019.34,425.431 2021.38,423.074 2023.43,420.713 2025.47,418.35 \n 2027.52,415.984 2029.57,413.616 2031.61,411.244 2033.66,408.87 2035.7,406.494 2037.75,404.114 2039.8,401.732 2041.84,399.347 2043.89,396.96 2045.93,394.569 \n 2047.98,392.176 2050.02,389.78 2052.07,387.382 2054.12,384.981 2056.16,382.577 2058.21,380.17 2060.25,377.761 2062.3,375.349 2064.35,372.934 2066.39,370.516 \n 2068.44,368.096 2070.48,365.673 2072.53,363.247 2074.58,360.819 2076.62,358.388 2078.67,355.954 2080.71,353.517 2082.76,351.078 2084.8,348.636 2086.85,346.191 \n 2088.9,343.744 2090.94,341.294 2092.99,338.841 2095.03,336.385 2097.08,333.927 2099.13,331.466 2101.17,329.002 2103.22,326.535 2105.26,324.066 2107.31,321.594 \n 2109.36,319.12 2111.4,316.642 2113.45,314.162 2115.49,311.679 2117.54,309.194 2119.59,306.705 2121.63,304.214 2123.68,301.721 2125.72,299.224 2127.77,296.725 \n 2129.81,294.223 2131.86,291.719 2133.91,289.211 2135.95,286.701 2138,284.188 2140.04,281.673 2142.09,279.155 2144.14,276.634 2146.18,274.11 2148.23,271.584 \n 2150.27,269.055 2152.32,266.523 2154.37,263.988 2156.41,261.451 2158.46,258.911 2160.5,256.368 2162.55,253.823 2164.59,251.275 2166.64,248.724 2168.69,246.17 \n 2170.73,243.614 2172.78,241.055 2174.82,238.493 2176.87,235.929 2178.92,233.362 2180.96,230.792 2183.01,228.219 2185.05,225.644 2187.1,223.066 2189.15,220.485 \n 2191.19,217.901 2193.24,215.315 2195.28,212.726 2197.33,210.135 2199.38,207.54 2201.42,204.943 2203.47,202.343 2205.51,199.741 2207.56,197.135 2209.6,194.527 \n 2211.65,191.917 2213.7,189.303 2215.74,186.687 2217.79,184.068 2219.83,181.447 2221.88,178.822 2223.93,176.195 2225.97,173.565 2228.02,170.933 2230.06,168.298 \n 2232.11,165.66 2234.16,163.019 2236.2,160.376 2238.25,157.73 2240.29,155.081 2242.34,152.429 2244.38,149.775 2246.43,147.118 2248.48,144.458 2250.52,141.796 \n 2252.57,139.131 2254.61,136.463 2256.66,133.792 2258.71,131.119 2260.75,128.443 2262.8,125.764 2264.84,123.083 2266.89,120.399 2268.94,117.712 2270.98,115.022 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},
"execution_count": 144,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@pipe range(0,1, length=100) |> plot(_, ecdf_H.(_), label=\"Empirical CDF (N=10000)\",\n",
" legend=:topleft)\n",
"@pipe range(0,1, length=100) |> plot!(_, ecdf_H_very_few.(_), label =\"Empirical CDF (N=20)\")\n",
"@pipe range(0,1, length=1000) |> plot!(_, H.(_), label =\"True CDF\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we can see, the procedure (with enough number of draws) succesfully replicates the discrete jump at $0.5.$"
]
}
],
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