Math 1780.001 (Cherry) Homework for Wednesday, July 29
Test #3. Test three will cover sections: 3.7, 3.8, 4.1, 4.2, 4.3, 4.4. You will not be required to do any “tricky”
integrals for the test, but you should be familiar with the basics of integration and know how to find anti-derivatives
for polynomial functions.
Reading. Read sections 4.1–4.4.
Practice Problems. Work out the solutions to the following problems in a notebook and check your answers in the
back of your text book. YOU WILL NOT TURN THESE PROBLEMS IN for a grade. They are only to help you
study. Note however that these problems may appear verbatim on the weekly tests.
4.7, 4.8, 4.9, 4.10, 4.11, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.31, 4.32, 4.33, 4.34, 4.35
Homework to turn in Wednesday, July 29 (at the start of class)
1. (10 points) Let Xbe the random variable measuring how long (in hours) a child can play a certain video game before
getting bored and wanting to do something else. Suppose that the probability distribution function p(x)for Xis of the
form:
p(x) =
0if x < 0
kx if 0≤x≤8
0if x > 8.
(a) Find k. Hint: Compute R∞
−∞ p(x)dx.
(b) Compute the cumulative disribution function F(x)and sketch its graph.
(c) What is the probability that the child will play from 1–4 hours before getting bored?
(d) Compute the expected amount of time a child will play before getting bored.
2. Do problem 4.20, but assume that the computer center is up for 90 minutes and then down for 30 minutes on a regular
cycle.
3. Do problem 4.21, but using a 10 hour day.
4. Suppose the volumes of spherical rain drops are uniformly distributed from 1 cubic milimeter to 10 cubic milimeters.
Find the mean and variance of the radii of the rain drops. Remember, that the volume vof a sphere of radius ris
v=4
3πr3.Hint: For purposes of this problem, I am using vto denote volume so as not to conflict with V, which we
have been using to denote “variance.” Apply Theorem 4.1.
5. Suppose that customers arrive at a checkout station according to a Poisson distribution with an average of 0.8customers
arriving per minute. (a) What is the expected time between customer arrivals? (b) If the checkout clerk needs 2 minutes
to process a customer, what is the probability another customer will arrive before the clerk finishes with the previous
customer?
