$\chi^2$ distribution

• The Chi squared distribution $\chi^2(k)$ with some integer $k$ degrees of freedom is the sum of squares of $k$ independent standard Normal variates.

using Distributions 
using Plots
gr()
using SpecialFunctions
using LaTeXStrings

@doc Chisq

Prepare standard normal distributions

d = Normal(0,1)

Plot histogram

sample = rand(d, 10000)
p = plot(xlim=[-3, 3])
histogram!(sample, normalize=:pdf, label="hist")
plot!(x->pdf(d, x),label="pdf")
savefig(joinpath(@OUTPUT, "normal.png"))

Prepare $\Gamma$ function

@doc gamma

Γ(x) = gamma(x);

function T(n, z) 
    if z > 0
        a = z^(n/2 - 1)*exp(-z/2)
        b = 2^(n/2) * Γ(n/2)
        return a/b
    else
        return 0.
    end
end;

Generate $\chi^2$ distribution with ...

degree = 1

n_trial = 100000
X = rand(d, n_trial)
X² = X .^ 2.
p = plot()
histogram!(p, X², normalize=:pdf, label=L"X^2")
x = range(0.01,2,length=100)
y = T.(1, x)
plot!(p, x, y, xlim=[0,2], ylim=[0,4], label=L"T_1")

y = pdf.(Chisq(1), x)
plot!(p, x, y, line=:dash, label=L"\chi^2(1)") ;

degree = 2

n_trial = 100000
X = rand(d, n_trial)
Y = rand(d, n_trial)
X² = X .^ 2
Y² = Y .^ 2
p=plot()

histogram!(p, X².+Y², normalize=:pdf, label=L"X^2+Y^2")
x = range(0.01,2,length=100)
y = T.(2, x)
plot!(p, x, y,xlim=[0,5], ylim=[0,0.5], label=L"T_3")

χ² = Chisq(2)

y = pdf.(χ², x)
χ² = Chisq(2)
plot!(p, x, y, label=L"\chi^2(2)",line=:dash)

degree = 3

n_trial = 100000
X = rand(d, n_trial)
Y = rand(d, n_trial)
Z = rand(d, n_trial)
X² = X .^ 2
Y² = Y .^ 2
Z² = Z .^ 2
p = plot()
histogram!(p, X².+Y².+Z², normalize=:pdf, label=L"X^2+Y^2+Z^2")
χ² = Chisq(3)

x = range(0.01,2,length=100)
y = T.(3, x)
plot!(p, x, y,xlim=[0,5], ylim=[0,0.5], label=L"T_3")
y = pdf.(χ², x)

plot!(p, x, y, label=L"\chi^2(3)",line=:dash)