Infinite Support
Here let us assume an endless box of chocolates with random selection with replacement of
Write out some of the sample space. Let \(F\) be the event of choosing a favorite chocolate, so \(F^{c}\) is the event of choosing an average chocolate, \(P(F) = p\) and \(P(F^{c}) = 1-p\). Let \(k\) be the amount of chocolates chosen reaching a favorite chocolate
Probability Mass Function
A geometric distribution is a discrete probability distribution with
- probability \(p\) for “success”
- probability \(1-p\) for “failure”
- for \(k = 0, 1, 2, 3, ...\)
- “success” on \((k+1)^{\text{th}}\) trial
The probability mass function (PMF) is
\[f(X = k) = (1-p)^{k}p\]
The cumulative mass function (CMF) is
\[F(X \leq k) = 1 - (1-p)^{k+1}\]
Cumulative Calculations
Ted Mosby is setting up his weekend plans. Barney convinced him to try out an app called Tinder. “Ted Mosby, Architect” has a 2/3 chance of setting up a date among those he “swipes right”. Suppose that Ted stops using Tinder this session once he sets up a date.
- Compute the probability that Ted needs exactly 4 ``swipes right’’ to set up a date.
- Compute the probability that Ted needs at most 4 ``swipes right’’ to set up a date.
Complementary Cumulative Mass Function
When modeling with a geometric distribution—i.e. \(X \sim Geo(p)\)—the probability that “success” takes more than \(k\) trials is
\[P(X > k) = (1-p)^{k+1}\]
When modeling with a geometric distribution—i.e. \(X \sim Geo(p)\)—the probability that “success” takes at least \(k\) trials is
\[P(X \geq k) = (1-p)^{k}\]
Rebecca Bunch follows Josh Chan, whom she briefly dated as a teenager, by moving to West Covina, California. Suppose that it may take a while for Rebecca and Josh to reunite and there is a 29 percent chance of them meeting during any particular week. What is the probability that it will take Rebecca more than 3 weeks to reunite with Josh?
What is the probability that it will take Rebecca at least 9 weeks to reunite with Josh given that she has already spent at least 5 weeks in West Covina?
Still holding on to that 29 percent chance of reuniting with Josh, compute the probability that it will take Rebecca at least 36 weeks to reunite with Josh given that she has already spent at least 32 weeks in West Covina.
Memoryless Property
The previous two examples demonstrated the memoryless property.
The geometric distribution is the only discrete probability distribution that has the memoryless property:
\[P(X \geq a + b | X \geq b) = P(X \geq a)\]
Looking Ahead
- due Fri., Feb. 10:
- WHW5
- JHW3
- Demographics Survey Part 2
- Be mindful of before-lecture quizzes
No lecture session for Math 32:
- Feb 20, Mar 10, Mar 24
Exam 1 will be on Wed., Mar. 1
- more information in weekly announcements
