Binomial distribution approximations

1 Introduction

2 Normal approximation

using Random, Distributions, CairoMakie

Random.seed!(1234)

# the probability of success
ps = [0.1, 0.3, 0.5, 0.9]
# the number of experiments
ns = [10, 100, 1000]
layouts = [[1, 1], [1, 2], [2, 1], [2, 2]]
colors = [:blue, :green, :red]
normalDist = Normal(0, 1)
N = 10^6

fig = Figure(size=(1400, 1200))
for i in 1:length(ps)
    p = ps[i]
    ax = Axis(fig[layouts[i][1], layouts[i][2]])
    totalBins = []
    for j in 1:length(ns)
        n = ns[j]
        binomialDist = Binomial(n, p)
        samples = rand(binomialDist, N)
        # normal standardization
        normSamples = @. (samples - n * p) / sqrt(n * p * (1 - p))
        bins = sort(unique(normSamples))
        stephist!(ax, normSamples; color=colors[j], normalization=:pdf, bins=bins, label=string("n = ", n))
        totalBins = vcat(totalBins, bins)
    end
    xGrid = round(minimum(totalBins), RoundDown; digits=0):0.01:round(maximum(totalBins), RoundUp; digits=0)
    lines!(ax, xGrid, pdf.(normalDist, xGrid); color=:black, label="N(0, 1)")
    axislegend(ax)
    ax.xlabel = string("x\n(p = ", p, ")")
    ax.ylabel = "Density"
end
fig

3 Poisson approximation

using Random, Distributions, CairoMakie

Random.seed!(1234)

lambda = 5
ns = [10, 20, 200]
N = 10^6
colors = [:red, :blue, :green]

fig = Figure()
ax = Axis(fig[1, 1]; xlabel="x", ylabel="Density")
poissonDist = Poisson(lambda)
samples = rand(poissonDist, N)
bins = sort(unique(samples))
stephist!(ax, samples; linewidth=3, linestyle=:dot, normalization=:pdf, bins=bins, color=:black, label="λ = $(lambda)")
for i in 1:length(ns)
    p = lambda / ns[i]
    binomialDist = Binomial(ns[i], p)
    samples = rand(binomialDist, N)
    bins = sort(unique(samples))
    stephist!(ax, samples; normalization=:pdf, bins=bins, color=colors[i], label="n = $(ns[i]), p = $(p)")
end
axislegend(ax)
fig