Introduction to Probability Models

Lecture 18

Qi Wang, Department of Statistics

Oct 2, 2017

Reminders

  1. The fourth quiz will be on this Wednesday, Oct 4
  2. The third homework is posted and due on Oct 16
  3. No class on Oct 6 as Exam 1 compensation

Example 1

Consider a game where we will roll a fair die. We will roll it until we get a $5$. What is the probability that it will take $3$ rolls to get the $5$?
Think about:

  • Are the subsequent rolls independent?
  • What about the probability of success on each roll?
  • Do we have a set number of trials?

GEOMETRIC RANDOM VARIABLE

Characteristics of the Geometric Distribution

  • The definition of $X$ : the number of trials to get the first success
  • Support: $\{1, 2, \cdots\}$, NOTE: NO ZERO!
  • Parameter: $p$, the probability of success in one trial
  • PMF: $P_X(x) = p(1-p)^{x - 1}$
  • Expected Value: $E[X] = \frac{1}{p}$
  • Variance: $Var(X) = \frac{1 - p}{p^2}$
  • $X \sim Geom(p)$

Example 2

Suppose Dunphy is really bad at tossing a Frisbee and unfortunate hits people walking by at a rate of 1 out of every 5 people.

  1. What is the probability that his first accidental hitting is the $6_{th}$ or $7_{th}$ person to walk by?
  2. What is the probability that more than 7 people walk past before he hits one with the Frisbee?

Important Properties for the Geometric Distribution

  • Tail Probability formula: $P(X > k) = (1 - p)^k$
  • Memoryless Property: $P(X > s + t| X > s) = P(X > t)$ and $P(X < s + t| X > s) = P(X < t)$

Example 2 continued

  • 3. Four people have walked past Dunphy and none have been hit by a Frisbee. What is the probability that at most 9 walk by until the first person is hit by a Frisbee?
  • 4. Four people have walked past Dunphy and none have been hit by a Frisbee. What is the probability that at least 10 walk by until the first person is hit by a Frisbee?

Example 3

Shaq is shooting free throws in the gym. He intends to stay until he makes one. His probability of making one on any free throw is 0.527. Let X be the number of attempts until he makes one.

  1. Distribution, parameter, support?
  2. Expected number of shots until he makes one?
  3. Probability he makes his first shot on the $4_{th}$ try?
  4. Probability it takes him at least 4 shots to make $1_{st}$?
  5. Probability it takes him exactly 4 shots if he already missed the first?
  6. Probability it takes him at least 4 shots if he already missed the first?