IEOR W4150 Introduction to Probability and Statistics Fall 2017
Dr. A. B. Dieker
Homework 5
due on Thursday October 12, 2017, 5:00pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.g., on wikipedia), then credit is only given for the intermediate steps.
1. Let U be a standard uniformly distributed random variable. (This means uniformly distributed
on the interval (0;1).)
(a) Calculate the distribution function of X = 2 log(U).
(b) What distribution does X have? What is (are) the parameter(s)?
2. Let U be a standard uniformly distributed random variable and N be a standard normally
distributed random variable. Show that, for any x, the random variable 1(U + (1 U) (x))
has the same distribution function as N given N > x. Here is the cumulative distribution
function of the standard normal distribution and 1 its inverse. (This fact can be exploited to
simulate normal random variables conditioned to exceed some level x.)
3. Suppose that Warren and Trump are the only presidential candidates in 2020. Let W and T be
random variables, where W stands for the number of Warren voters and T for the number of
Trump voters (in millions). The joint probabilities P(W = k;T = ‘) are given in the following
table.
k
75 125
‘ 60 2/10 1/10
100 3/10 4/10
(a) Compute the probability that Warren gets more votes than Trump.
(b) Find the probability mass function p of the total number of voters W + T.
(c) Compute the probability that there are more voters than in 2016, assuming that there were
150 million voters then.
4. Let X and Y be discrete random variables. The joint probabilities P(X = k;Y = ‘) are given
in the following table.
k
0 5 10
‘ 0 1/21 2/21 3/21
5 4/21 5/21 6/21
(a) Let F be the joint distribution function of X and Y . Calculate F(6;5) and F(5;4).
(b) Calculate the marginal probability mass function of X and Y .
(c) Compute the probability mass function of X 2Y .
(d) Compute the probability mass function of XY .
5. Suppose X and Y have a joint pdf f given by
f(x;y) =
(
ce y+x2 1=2 x 1=2;y > x2;
0 otherwise.
(a) Determine the value of c.
(b) Calculate the marginal probability density function of X.
(c) Only use distribution functions in your answer to the following question: are X and Y
independent? (Wait with this one until after the lecture on 10/9).
6. Let X and Y be two continuous random variables for which the joint distribution function
satis es F(x;y) = 16xy for 0 x 2;0 y 3.
(a) Determine the values the vector (X;Y ) can take.
(b) Specify F(x;y) for (x;y) outside the ’rectangle’ f(x;y) : 0 x 2;0 y 3g.
7. Let X and Y be two continuous random variables with joint probability density function
f(x;y) =
(
Kx 0 x 1;0 y 1
0 otherwise;
for some constant K > 0.
(a) Determine K.
(b) Find the probability P(1=4 X 3=4;1=3 Y 2=3).
(c) Calculate the distribution function of Y .
8. Suppose that the joint distribution function of X and Y is given by
F(a;b) = 1 4 bac 2 bbc + 2 2bac bbcif a > 0 andb > 0;
and F(a;b) = 0 otherwise. Here bxc is the largest integer smaller or equal to x (‘rounding
down’).
(a) Determine the marginal distribution functions of X and Y .
(b) Determine the marginal probability mass functions of X and Y . Do you recognize one of
the standard probability distributions?
(c) Determine the joint probability mass function of X and Y .
(d) Are X and Y independent? (Wait with this one until after the lecture on 10/9).
9. Assume both the thickness and the hole diameter of a washer vary from washer to washer. Let X
and Y denote the thickness and hole diameter, respectively, in mm. Assume the joint probability
density function f of X and Y is given by
f(x;y) =
(1
6(x + y) 1 x 2;4 y 5;
0 otherwise.
Are X and Y independent? (Wait with this one until after the lecture on 10/9).
2
10. Two random variables X and Y have a joint probability density function f given by
f(x;y) =
(
24xy if x 0;y 0;x + y 1
0 otherwise
(a) Verify that f is a joint probability density function.
(b) Calculate the marginal probability density function of X.
(c) Calculate P(X 1=4;Y 1=4).
(d) Are X and Y independent? (Wait with this one until after the lecture on 10/9).
11. Suppose that U1, U2, and U3 are independent random variables with the standard uniform
distribution. Show that X = min(U1;U2;U3) has probability density function f(x) = 3(1 x)2,
x2(0;1) and zero otherwise. (Hint: rst calculate the distribution function of X.) (Wait with
this one until after the lecture on 10/9).