## Introduction to Probability Models

Lecture 13

Qi Wang, Department of Statistics

Sep 19, 2018

## Revision

**Permutation: **Ordered arrangement of r distinct objects from a set of n objects. $$_nP_r = P_r^n = \frac{n!}{(n - r)!}$$
**Combination:**Unordered arrangement of r distinct objects from a set of n objects. $$_nC_r = C_r^n = \frac{n!}{(n - r)! r!}$$
**Multinomial Coefficient:**m objects are in k distinct groups, size of groups are $m_1, m_2, \cdots, m_k$, number of ways to order these are: $$\binom{m}{m_1, m_2, \cdots, m_k} = \frac{m!}{m_1!m_2!\cdots m_k!}$$

## Properties of Expected Value

X, Y are random variables, c and d are constant

- $E[c] = c$
- $E[cX] = cE[X]$
- $E[X + Y] = E[X] + E[Y]$
- $E[cX + dY] = cE[X] + dE[Y]$

## Properties of Variance

X, Y are random variables, c and d are constant

- $Var(X) = E[(X - E[X])^2] = E[X^2]-E[X]^2$
- $Var(c) = 0$
- $Var(cX) = c^2 Var(X)$
- If X and Y are independent, $Var(X + Y) = Var(X) + Var(Y)$
- If X and Y are independent, $Var(cX + dY) = c^2Var(X) + d^2Var(Y)$

## Example 1

Suppose X and Y are random variables with $E[X] = 3, E[Y] = 4$ and $Var(X) = 2, Var(Y) = 1$. Find

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