Math 309 - Christian Brothers University
Math 309 Bivariate Continuous Distributions
1. Let the joint probability density function of random variables X and Y be given by
[pic]
a) Find [pic]
b) Find [pic].
c) Find the marginal density functions of X and Y respectively.
d) Are X & Y independent? Justify your response.
e) Find E[X], E[Y], E[XY], and E[X-Y].
2. Let the joint probability density function of random variables X and Y be given by
[pic]
a) Find P(X < ¾, Y < ¼).
b) Find P(X < ¾, Y < ½).
c) Find P(X < ¾ | Y < ½).
3. Consider the joint density function: [pic]
a) Find [pic]
b) Find [pic]
c) Find the marginal density functions of X and Y respectively.
d) Are X & Y independent? Justify your response.
e) Find E[X], E[Y], and E[XY].
4. Consider the joint density function: [pic]
a) Find [pic].
b) Find [pic]
5. Let X & Y denote the proportions of time, out of one work week, that employees I and II respectively, actually spend performing their assigned tasks. The joint relative frequency behavior is modeled by: [pic]
a) Find [pic].
b) Find [pic].
c) Find [pic].
d) Interpret a-c in the context of the problem.
e) Are X & Y independent? Justify your response.
6. A man invites his fiancée to an elegant hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 a.m. and 12 noon. Let X denote the time that the man arrives and Y denote the time that his fiancée arrives. If they arrive at random times during this period, what is the probability that the first to arrive has to wait at least 12 minutes? Find the solution using the joint distribution of X and Y.
7. X and Y possess the following joint density function.
[pic]
a) P( X > 2Y)
b) P( X > 1/4, Y < 3/4 ) (Hint: sketch the domain!)
8. X and Y possess the following joint density function.
[pic]
a) Find the marginal density functions for X and Y.
b) Find P( X > 1/2 ( Y > 1/4 ).
c) Are X and Y independent? Justify your response.
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