HELP PROBABILITY

Question

5. The random variables X, Y have joint probability density function
f(x, y) = (
C
e
−x−e
−x−2y
e
y−1
if x > 0 and y > 0,
0 otherwise.
(a) What is the value of C?
(b) Are X and Y independent?
(c) Find P(X < Y ).

Answer 1

Hi JJ! Let's walk through how to solve these problems.

 

(a) You know the density function, but to solve for c you need an equation--some more information about the density function. We know that ALL density functions integrate to 1, correct? Because the probability that X and Y equals ANYTHING is one. Try integrating the density across all possible values of X and Y, set the result equal to 1, and then solve for c.

 

(b) ONE WAY to determine independence is to find the conditional distribution of Y given X. To find the conditional distribution of Y given X, divide the joint distribution of Y and X by the marginal distribution of X. If you can't simplify the result to get rid of the Xs, then X and Y are not independent.

 

(c) To find this probability, we simply need to integrate the density function over the set of all (x,y) values where X<Y.

 

Please let me know if you would like a little more detail on how to go through these problems. We can set up a tutoring session together!