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Ch04.jl
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Ch04.jl
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# Chapter 4.3
# CDF of a Gaussian mixture
## Julia code to generate the PDF and CDF
using Distributions
using Plots
gmm = MixtureModel(Normal[Normal(0, 1), Normal(5, 1),], [0.3, 0.7])
p1 = plot(x -> 0, -5, 1, fillrange= x->pdf(gmm, x),
linealpha=0, fillcolor=RGB(0.8, 0.8, 1), alpha=0.5, label=false) # style
plot!(p1, x -> pdf(gmm, x), -5, 10,
color=1, linewidth=6, label="PDF") # style
p2 = plot(x -> cdf(gmm, x), -5, 10,
linewidth=6, label="CDF")
plot(p1, p2, layout=(1, 2),
size=(600, 300), legend=:topleft)
# CDF of a uniform random variable
# Julia code to generate the PDF and CDF of a uniform random variable
using Distributions
using Plots
u = Uniform(-3, 4)
# pdf
p1 = plot(x -> 0, -5:0.01:1, fillrange=x->pdf(u, x),
linealpha=0, fillcolor=RGB(0.8, 0.8, 1), alpha=0.5, label=false)
plot!(p1, x -> pdf(u, x), -5, 10,
color=1, linewidth=6, ylims=(0, 0.4), label="PDF")
# cdf
p2 = plot(x -> cdf(u, x), -5:0.01:10,
linewidth=6, label="CDF")
vline!(p2, [-3, 4],
linestyle=:dash, color=:green, label=false)
plot(p1, p2, layout=(1, 2),
size=(600, 300), legend=:topleft)
# CDF of an exponential random variable
# Julia code to generate the PDF and CDF of an exponential random variable
using Distributions
using Plots
u = Exponential(2)
# pdf
p1 = plot(x -> 0, -5:0.01:1, fillrange=x->pdf(u, x),
linealpha=0, fillcolor=RGB(0.8, 0.8, 1), alpha=0.5, label=false)
plot!(p1, x -> pdf(u, x), -5, 10,
color=1, linewidth=6, ylims=(0, 0.6), label="PDF")
# cdf
p2 = plot(x -> cdf(u, x), -5:0.01:10,
linewidth=6, label="CDF")
plot(p1, p2, layout=(1, 2),
size=(600, 300), legend=:topleft)
# Chapter 4.5
# Generate a uniform random variable
# Julia code to generate 1000 uniform random numbers
using Distributions
using Plots
u = Uniform(0, 1) # same as Uniform()
X = rand(u, 1000)
histogram(X)
# Mean, variance, median, mode of a uniform random variable
# Julia code to compute empirical mean, var, median, mode
using Distributions
u = Uniform(0, 1)
X = rand(u, 1000)
M = mean(X)
V = var(X)
Med = median(X)
Mod = mode(X)
# Probability of a uniform random variable
# Julia code to compute the probability P(0.2 < X < 0.3)
using Distributions
u = Uniform(0, 1)
F = cdf(u, 0.3) - cdf(u, 0.2)
# PDF of an exponential random variable
# Julia code to generate PDF and CDF of an exponential random variable
using Distributions
using Plots
exp1 = Exponential(1/2)
exp2 = Exponential(1/5)
# Plotting the pdfs
p1 = plot(x -> pdf(exp1, x), 0, 1,
linewidth=4, linestyle=:dashdot, color=RGB(0,0.2,0.8), label="λ = 2")
plot!(p1, x -> pdf(exp2, x), 0, 1,
linewidth=4, color=RGB(0.8,0.2,0), label="λ = 5")
# Plotting the cdfs
p2 = plot(x -> cdf(exp1, x), 0, 1,
linewidth=4, linestyle=:dashdot, color=RGB(0,0.2,0.8), label="λ = 2", legend=:topleft)
plot!(p2, x -> cdf(exp2, x), 0, 1,
linewidth=4, color=RGB(0.8,0.2,0), label="λ = 5")
# Figure with both plots
plot(p1, p2, layout=(1, 2),
size=(600, 400))
# Chapter 4.6
# PDF and CDF of a Gaussian random variable
# Julia code to generate standard Gaussian PDF and CDF
using Distributions
using Plots
normdist = Normal()
p1 = plot(x -> 0, -5:0.01:-1, fillrange=x->pdf(normdist, x),
linealpha=0, fillcolor=RGB(0.8, 0.8, 1), alpha=0.5, label=false)
plot!(p1, x -> pdf(normdist, x), -5, 5,
color=1, linewidth=6, ylims=(0, 0.6), label="PDF")
p2 = plot(x -> cdf(normdist, x), -5:0.01:5,
linewidth=6, label="CDF")
plot(p1, p2, layout=(1, 2),
size=(600, 300), legend=:topleft)
# Skewness and kurtosis of a random variable
# Julia code to plot a Gamma distribution
using Distributions
using Plots
θ = 1
p1 = plot(size=(600, 300))
plot!(p1, x -> pdf(Gamma(2, θ), x), 0, 30, lw=4, c=RGB(0,0,0), label="k = 2")
plot!(p1, x -> pdf(Gamma(5, θ), x), 0, 30, lw=4, c=RGB(0.2,0.2,0.2), label="k = 5")
plot!(p1, x -> pdf(Gamma(10, θ), x), 0, 30, lw=4, c=RGB(0.4,0.4,0.4), label="k = 10")
plot!(p1, x -> pdf(Gamma(15, θ), x), 0, 30, lw=4, c=RGB(0.6,0.6,0.6), label="k = 15")
plot!(p1, x -> pdf(Gamma(20, θ), x), 0, 30, lw=4, c=RGB(0.8,0.8,0.8), label="k = 20")
# Julia code to compute skewness and kurtosis
X = rand(Gamma(3, 5), 10_000)
s = skewness(X)
k = kurtosis(X)
# Chapter 4.8
# Generating Gaussians from uniform
# Julia code to generate Gaussian from uniform
using Distributions
using Plots
mu = 3
sigma = 2
U = rand(10_000)
gU = sigma * quantile(Normal(), U) .+ mu
p1 = histogram(U, label="U")
p2 = histogram(gU, label="gU")
plot(p1, p2, layout=(1, 2), size=(600, 300), legend=:topleft)