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?
Solution
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.
02
Insurance expected payout
A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between $100 and $1500. The manager wants to offer customers a policy that has a $500 deductible and covers all costs above the deductible.
How much is the expected payout per accident?
(Hint: Graph the PDF for the cost of repairs
; write a formula for the payout in terms of using cases; then integrate.)
Solution
06
(1) Find PDF of
. If
, insurance covers 0$. If
, then the insurance covers dollars.
(2) Integrate to find
. Since the cost of repairs in uniformly distributed, we have
, . Link to original
03
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.)
Solution
11
(1) State the PMF of a geometric random variable.
(2) Use formula for expectation to find
.
(3) Apply hint.
We have that
. Differentiating both sides yields
. Note that here,
, so
(4) Find expression
, and note that . Applying the hint, we have
. Using the linearity of expectation, we can write this as
.
(5) Find
and First, note that the second derivative of
is . Thus,
.
(6) Find
. Link to original
04
Tutoring needs
A course with 6 students offers free one-on-one tutoring to each student for 1 hour the week before the final exam. One tutor, Jim, has been hired to provide this tutoring, but he is available for only 4 hours that week. The instructor of the course will tutor any students that Jim is not able to help. Jim will be paid $20 per hour by the department. The instructor will provide tutoring for free. Let
be the number of students that will need tutoring. The PMF of is given below. (a) Find the probability the instructor will need to provide tutoring.
(b) Find the expected value of the number of students that will need tutoring.
Solution
13
(a)
(b)
Link to original
05
Expectation from CDF
The CDF of random variable X is given by:
Compute
.
Solution
16
(a)
(b)
Link to original