Suppose x and y are random variables with joint density function. f(x, y) = 0.1e−(0.5x + 0.2y) if x ≥ 0, y ≥ 0 0 otherwise (a) is f a joint density function? yes no (b) find p(y ≥ 8). (round your answer to four decimal places.) find p(x ≤ 5, y ≤ 8). (round your answer to four decimal places.) (c) find the expected value of x. find the expected value of y.

a. is a joint density function if its integral over the given support is 1:

so the answer is yes.

b. We should first find the density of the marginal distribution, :

Then

or about 0.2019.

For the other probability, we can use the joint PDF directly:

which is about 0.7326.

c. We already know the PDF for , so we just integrate:

the graph on the bottom right

step-by-step explanation: