## Introduction to Probability Models

Lecture 12

Qi Wang, Department of Statistics

Sep 17, 2018

### Expected Value of A Discrete Random Variable

**Definition:** Weighted average of the possible values,
$$E[X] = \sum_x x \times p_X(x)$$
- Expected value can be positive or negative
- It does
**NOT** have to be an integer

### Some Properties of Expected Value

- c is a constant, $E[cX] = cE[X]$
- $E[X + Y] = E[X] + E[Y]$

## Example 1

$X \sim p_X(x) = P(X = x) = k(5 - x), x \in \{0, 1, 2, 3, 4\}$, if X
has the valid pmf, find the expected value of X

### Variance of A Discrete Random Variable

**Definition:**measures of spread, relates how far a particular value of the r.v. is from the average
(i.e. expected value) of the r.v $$Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$$
- The variance will NEVER be negative.
**Standard Deviation:** Square root of the variance $SD(X) = \sqrt{Var(X)}$

### Some Properties of Variance

- c is a constant, $Var(cX) = c^2 Var(X)$
- If X and Y are independent, $Var(X + Y) = Var(X) + Var(Y)$

### Example 2

For the unfair coin problem in Lecture 11, find $E[3X - 2]$, $SD[3X - 2]$

### Example 3

Suppose X and Y are random variables with $E[X] = 3, E[Y] = 4$ and $Var(X) = 2$. Find

- $E[2X + 1]$
- $E[X – Y] $
- $E[X^2]$
- $E[X^2 – 4]$
- $E[(X – 4)^2]$
- $Var(2x – 4)$