# Week 3

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

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.

##### PMF

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 $$y$$.

##### CDF

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

• $$E\{cg(Y)\}=cE\{g(Y)\}$$

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

#### Variance

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

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

## Resources

### First Lecture

Slides Videos
Slides Video 001 Video 002

### Second Lecture

Slides Videos
Slides Video 001 Video 002