In these problems,
Bernoulli process
01
[duplicated W03-08]
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.
03
Rolling until a six
A fair die is rolled until a six comes up.
What are the odds that it takes at least 10 rolls? Use a geometric random variable.
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 after day 4 and by day 10?
(You have calculated the answer before. This time, rework the problem in terms of an appropriate distribution type.)
05
A very strange car A very strange car with
components will drive if at least half of its components work. Each component will work with the same probability , independently of the others. For what values of
is a car with more likely to drive than a car with ? (Start by defining a random variable that counts the number of working components.)
06
Geometric distribution is memoryless
Suppose that
. Derive this equation:
Interpret the equation. (Inspired by the title.)
07
Binomial ratios Suppose
.
- Find the value of
that maximizes . Do this by studying the successive ratios . - Use these ratios to compute
as a sum of 5 terms without using factorials. Do this by computing directly, and then writing a recursive algorithm that determines in terms of .
08
Prize on the Mall
A booth on the Mall is running a secret prize game, in which the
passerby wearing a hat wins $1,000. Passersby wear hats independently of each other and with probability 20%.
Let
be a random variable counting how many passersby pass by before a winner is found. (a) What is the name of the distribution for
? What are the parameters? (b) What is the probability that the
passerby wins the prize? (c) What is the probability that at least
passersby are needed before a winner is found?
Expectation and variance
09
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?
10
Insurance expected payout
A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between
1500. The manager wants to offer customers a policy that has a $500 deductible and covers all costs above the deductible. How much is the expected payout per accident?
(Hint: Graph the PDF for the cost of repairs
; write a formula for the payout in terms of using cases; then integrate.)
11
Expectation, variance of geometric variable Derive formulas for
and given . Hint: For
you will get a sum that has terms like .
This series comes from the geometric series
(Differentiate both sides.)For
you will need to consider this general fact of algebra: (And apply the same methods as above.)
Poisson process
[All on this topic are in HW for W05.]