Question: 1. Nine IUSB students are going on a trip to Indianapolis for a Clots game. Go Colts!
(a) They will seat in a row. In how many ways can they seat if Josh, Mark and Mary will sit together?
(b) Four of them are juniors and the rest are seniors. If three students will be selected as designated drivers, what is the probability that at most one junior is selected?
(c) They will travel with three cars that hold 2, 4, and 5 passengers, respectively. All of them can drive. Find the number of possible ways to allocate 9 people to 3 cars that each car has at least one person (each car has to have a driver) if
(i) the allocations are to the cars only (disregard the seats in the car).
(ii) the allocations are to the seats.
Question: 2. Draws in the problem are without replacement.
There are
9 clocks in Box 1, 4 of them made by company A, 2 by company B and 3 by company C;
8 clocks in Box 2, 2 of them made by company A, 1 by company B and 5 by company C;
6 clocks in Box 3, 1 of them made by company A, 3 by company B and 2 by company C.
(a) Draw 4 clocks from Box 1, Find the probability that at least one clock from each company.
(b) Draw 2 clocks from Box 3. Let Xi = number of clocks from company A, B and C, respectively for i =1, 2, 3. Let Y1 = X1 and Y2 = X2 - X3. Find the joint probability function of Yi and Y2.
(c) You toss a coin two times. If no head is observed, you draw 3 clocks from Box 1. If two heads are observed, you draw 3 clocks from Box 2. Otherwise, you draw 3 clocks from Box 3.
Given that you would get 2 clocks from company C and 1 clock from company B, find the probability that theses clocks are from Box 2.
Question: 3. Only a normal table can be used to facilitate computation for this problem (no normalcdf from a calculator).
Susan commutes daily from her home to her office. The average time for a one-way trip is 24 minutes with a standard deviation of 3.8 minutes. Assume that the trip time follows a normal distribution.
(a) If office opens at 9:00 am and she leaves her house at 8:35 am daily, what percentage of the days that she would be late for work?
(b) If she leaves her house at 8:30 am and coffee is served at the office from 8:50 am until 9:00 am, what is the probability that coffee is not served at the time she gets her office?
(c) Find the length of time below which we find the fastest 15% of the trips.
(d) To have 99% of days not being late for work, what is the latest time she must leave home?
(e) Find the probability (by formula) that at most 2 of the next 20 trips will take at least 1/2 hour.
(f) A trip to a client's office from her home takes 30 more minutes than twice the time to her own office.
Let W be the time for a trip to the client's office.
(i) Find mean, variance and moment generating function of W. Identify the distribution of W.
(ii) Find the probability that a trip to the client's office takes more than 1 hour but less than 1.5 hour.
Question: 4. Use formulas to carry out all computations in this problem.
The number of calls received at a 24-hour service desk follows a Poisson distribution that has a standard deviation
(a) Find the probability that the service desk will receive more than 3 calls in the next 5 minutes.
(b) Let Y be the number of calls that the service desk will receive in the next 10 minutes. Find the moment generating function of Y.
(c) Find the probability that the time between the last call and the next call will be between 1 to 4 minutes.
(d) Rick starts his new job at the service desk today. Find the probability that his fifteenth minute will be the fifth minute that the service desk receives exact one call.
(e) Let X be the time until he receives 3rd call. Find the P(X < 5).
Question: 5. A function F is defined as the following:
0, if y < -Π/2;
a.cos y, if -Π/2 ≤ y < 0;
F(y) = b, if 0 ≤ y < Π/2;
1 - c.sin y, if Π/2 ≤ y < Π;
2c, if y ≥ 3.
(a) Find constants a, b and c such that F is the cdf of a continuous random variable Y. Sketch a graph of F(y).
(b) Find the pdf of Y and sketch a graph of the pdf.
(c) Find (|Y| > 0.75Π).
(d) Find E(Y2)
(e) Find an interval that contains exact 75% of the probability.
Question: 6. Two continuous random variables Y1 and Y2 has the joint density function
f (y1, y2) = c(y1 + y2), if y1 > 0, y2 > 0, y1 + y2 < 2)
= 0, otherwise
(a) Find the constant c.
(b) Find P(Y1 > 1/2|Y2 < 1/4).
(c) Are Y1 and Y2 independent? Verify your answer.
(d) Find the conditional pdf f(y1|y2). Specify the region where the density is positive.
(e) Find P(Y1 > 0.5|Y2 = 0.5).
(f) Find Cov(Y1, Y2).