Introduction to Probability Models

Lecture 12

Qi Wang, Department of Statistics

Sep 17, 2018

Expectation

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

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

1. $E[2X + 1]$
2. $E[X – Y]$
3. $E[X^2]$
4. $E[X^2 – 4]$
5. $E[(X – 4)^2]$
6. $Var(2x – 4)$