Introduction to Probability Models

Lecture 12

Qi Wang, Department of Statistics

Sep 18, 2017

Reminders

  1. The third quiz will be on this Wednesday
  2. The second homework is due on this Friday, September 22
  3. The first exam will be from 8:00pm to 9:30pm next Tuesday, Sep 26

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[X2 – 4]
  5. E[(X – 4)^2]
  6. Var(2x – 4)