Week 3

This week, we will discuss the properties of discrete random variables.

February 6, 2023

Learning Outcomes

First Lecture

  • Describe Discrete Distributions

Second Lecture

  • Expected Values

  • Variance

  • Properties of Expected Values

Important Concepts

First Lecture

Discrete Variables

A random variable is considered to be discrete if it can only map to a finite or countably infinite number of distinct values.


The probability mass function of discrete variable can be represented by a formula, table, or a graph. The Probability of a random variable Y can be expressed as \(P(Y=y)\) for all values of \(y\).


The cumulative distribution function provides the \(P(Y\leq y)\) for a random variable \(Y\).

Commonly Used Distributions
Distribution Parameter(s) PMF \(P(Y=y)\)
Bernoulli \(p\) \(p\)
Binomial \(n\) and \(p\) \((^n_y)p^y(1-p)^{n-y}\)
Geometric \(p\) \((1-p)^{y-1}p\)
Negative Binomial \(r\) and \(p\) \((^{y-1}_{r-1})p^{r-1}(1-p)^{y-r}\)
Hypergeometric \(N\), \(n\), and \(r\) \(\frac{(^r_y)(^{N-r}_{n-y})}{(^N_n)}\)
Poisson \(\lambda\) \(\frac{\lambda^y}{y!} e^{-\lambda}\)

Second Lecture

Expected Value

The expected value is the value we expect when we randomly sample from population that follows a specific distribution. The expected value of a discrete random variable is \(Y\) with PMF of \(P(y)\) is

\[ E(Y)=\sum_y yP(y) \]

Expected Value Properties

Special properties of the expected value:

  • \(E(c)=c\), where \(c\) is constant

  • \(E\{cg(Y)\}=cE\{g(Y)\}\)

  • \(E\{g_1(Y)+\cdots+g_n(Y)\}=E\{g_1(Y)\}+\cdots+E\{g_n(Y)\}\)


The variance represents the variation of a random variable. The variance for a random variable Y is

\[ Var(Y) = E[\{Y-E(Y)\}^2] \]


First Lecture

Slides Videos
Slides Video 001 Video 002

Second Lecture

Slides Videos
Slides Video 001 Video 002