Week 3
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 |