Introduction to Probability Models

Lecture 13

Qi Wang, Department of Statistics

Sep 20, 2017

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: $$\binom{m}{m_1, m_2, \cdots, m_k} = \frac{m!}{m_1!m_2!\cdots m_k!}$$

Time for Quiz

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 $Var(X) = 2$. Find

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