Introduction to Probability Models

Lecture 15

Qi Wang, Department of Statistics

Sep 24, 201

Reminders

  • The first exam will be at CL50(Class of 1950 Lecture Hall) from 8:00pm to 9:30pm on Tuesday, Sep 25

Concepts

  • Random Experiment:
  • Element
  • Event
    • $\emptyset$
    • $S$ or $\Omega$
  • Union $\cup$:
  • Intersection $\cap$:
  • DeMorgan's Law:
  • Mutually Exclusive, Exhaustive and Partition

Concepts

  • Conditional Probability: if event $B$ has nonzero probability ($P(B) > 0$), $P(A|B) = \frac{P(A\cap B)}{P(B)}$
  • Multiplication Rule: $P(A\cap B) =P(B) \times P(A|B)$
  • General addition rule:
    • $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
    • $P(A\cup B \cup C)$
  • Independence:
    • \(P(A|B) = P(A)\)
    • \(P(B|A) = P(B)\)
    • \(P(A\cap B) = P(A)\times P(B)\)
    • Mutually exclusive events are NOT independent unless one of them has zero probability
    • Mutually Independent and Pairwise Independent
  • Law of Total Probability:$P(A) = \sum_{i = 1}^n P(A|B_i) \times P(B_i)$
  • Bayes Rule: $P(B_i|A) = \frac{P(B_i\cap A)}{\sum_{i = 1}^n P(A|B_i) \times P(B_i)}$

Concepts

  • Basic Counting Rules
  • Permutation: $_nP_r = P_r^n = \frac{n!}{(n - r)!}$
  • Combination: $_nC_r = C_r^n = \frac{n!}{(n - r)! r!}$
  • Multinomial Coefficient: $\binom{m}{m_1, m_2, \cdots, m_k} = \frac{m!}{m_1!m_2!\cdots m_k!}$
  • Random Variable
  • Probability Mass Function
    1. For every x, $0 \le p_X(x) \le 1$
    2. $\sum_x{p_X(x)} = 1$
  • Expected Value: $E[X] = \sum_x x \times p_X(x)$
    • c is a constant, $E[cX] = cE[X]$
    • $E[X + Y] = E[X] + E[Y]$
  • Variance: $Var(X) = E[(X - E[X])^2] = E[X^2]-E[X]^2$
    • $Var(cX) = c^2 Var(X)$
    • If X and Y are independent, $Var(X + Y) = Var(X) + Var(Y)$

Two Diagrams

  1. Venn Diagram
  2. Tree Diagram

Examples