Random Variables

• Example1 Assume a sample space $\Omega$ which consists of six names. Assume that probability function, $\mathbb{P}(\cdot)$, assigns uniform probability to each of the names. Let now, $X: \Omega \to \mathbb{Z}$, be the function that counts the number of letters in each name. Then question is then finding: $$p(x) := \mathbb{P}(X=x), \quad \text{for } x\in \mathbb{Z}$$
• The function $p(x)$ represents the probability distribution of the random variable $X$. Since $X$ measures the length of names, $X$ is a discrete random variable, and its probability distribution may be represented by Probability Mass Function (PMF), such as $p(x)$
• Simulation by Julia

using StatsBase, Plots

names = ["Mary", "Mel", "David", "John", "Kaylay", "Anderson"]
randomName() = rand(names)
X = 3:8
N = 10^6
sampleLengths = [length(randomName()) for _ in 1:N]

bar(X, counts(sampleLengths)/N, ylims=(0,0.35),
    xlabel="Name Length", ylabel="Estimated p(x)", legend=:none)

Types of Random Variables

• Discrete Random Variable i.e. The number of letters of one's name.
• Continuous Random Variable i.e. The weights of people randomly selected from a big population.
• In describing the probability distribution of a continuous random variable, the probability mass function, $p(x)$, is no longer applicable. This is because for a continuous random variable $X$, $\mathbb{P}(X = x)$ for any particular value of $x$ is $0$. Hence, in this case, the Probability Density Function (PDF), $f(x)$ is used, where $$f(x)\Delta\approx\mathbb{P}(x\le X \le x+\Delta)$$
• Here the approximate becomes exact as $\Delta\to 0$
• Three examples of distributions
  • Discrete Distribution: $$p(x)\left\{\begin{array}{ll}0.25 & \text{for }x=0,\\0.25 & \text{for }x=1,\\ 0.5 & \text{for }x=2.\end{array} \right. \\ \Downarrow\\ \sum_{x}p(x) = 1$$
  • Continuous Distribution: $$f_{1}(x) = \frac{3}{4}(1-x^{2})\quad \text{for } -1\le x \le 1 \\\\ f_{2}(x) = \left\{\begin{array}{ll} x + 1 & \text{for }x\in[-1,0],\\ 1 - x & \text{for }x\in(0,1]. \end{array} \right.\\ \Downarrow\\ \int_{-\infty}^{\infty}f_{i}(x)dx = 1 \quad \text{for } i = 1,2$$
• Simulation by Julia

using Plots, Measures;pyplot()

pDiscrete = [0.25, 0.25, 0.5]
xGridD = 0:2

pContinous(x) = 3/4*(1-x^2)
xGridC = -1:0.01:1

pContinous2(x) = x < 0 ? x+1 : 1-x

p1 = plot(xGridD, line=:stem, pDiscrete, marker=:circle, c=:blue, ms=6, msw=0)
p2 = plot(xGridC, pContinous.(xGridC), c=:blue)
p3 = plot(xGridC, pContinous2.(xGridC),c=:blue)

plot(p1,p2,p3, layout=(1,3), legend=false, ylims=(0,1.1), xlabel="x", 
    ylabel=["Probability" "Density" "Density"], size=(1200,400), margin=5mm)