Due date: Friday 9/19, 9:00am
Bernoulli process
01
02
PMF and CDF: number of heads in five flips
Let
count the number of heads resulting from five flips of a coin. Write complete formulas (using cases) for the PMF and CDF of
. Link to originalSolution
02
(1) Identify distribution.
We have that
, since there are trials and .
(2) Use binomial distribution formula to get PMF.
(3) Find CDF.
Since
, we have
(4) Write out explicit values of CDF.
Link to original
02
04
Intersection accidents
Suppose that the odds of an accident occurring on any given day at the intersection of Ivy and Emmet is 0.05.
What are the odds of the first accident occurring between day 5 and day 10, inclusive? (Use an appropriate discrete distribution type.)
Link to originalSolution
04
(1) Define random variables.
Let
. We wish to find
.
(2) Compute
using the formula for a geometric distribution. Link to original
03
09
Lottery game
Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.
If you keep purchasing these game cards until you win twice, what is the probability you will purchase at least 4 of them?
Link to originalSolution
Expectation and variance
04
01
Students and buses expect different crowding
Bus One has 10 students, Bus Two has 20, Bus Three has 30, and Bus Four has 40.
- Let
measure the number of students on a given random student’s bus. - Let
measure the number of students on a given random driver’s bus. Compute
and . Are they different? Why or why not? Link to originalSolution
09
(1) Compute
. There are
total students. Let
be the probability that Bus is selected, where . Note that
.
(2) Compute
. Let
be the probability that Bus is selected. Since it’s based off the drivers,
for all .
(3) Interpret solution.
Link to original
, because the probability that bus was selected in both scenarios varied.
05
06
Variance from CDF: Drill bit changes
The bits for a particular kind of drill must be changed fairly often. Let
denote the number of holes that can be drilled with one bit. The CDF of is given below: (a) Find the probability that a bit will be able to drill more than 2 holes.
(b) Find
by constructing the PMF. Link to originalSolution
10
(a)
(b)
Link to original
Review problems
06
04
Reliability - Math competition cutoff score
At a high school math competition, students take a test with 10 questions. Each question is worth one point and the probability of a student getting any one question correct is 0.55, independent of the other questions.
(a) Find the probability of a student getting a score of 8 or higher.
(b) Students take the test individually but compete in teams of 2. To proceed to the second round of competition, each student on the team must score at least 8. Each high school can enter 2 teams. If a high school enters two teams, find the probability at least one of their teams will make it to the second round. Assume students’ scores are independent.
Link to originalSolution
16
(a)
(b)
Link to original