Question

Let X and Y have joint PDF

f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y < infinity

with parameters λ > 0 and μ > 0

a) Find c such that this is a PDF.

b) Show that X and Y are Independent

c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in exponential form

d) Find the marginal distribution of Y, f(y)

e) Find E(Y)

Answer 1

(a) We need to find c, such that,

becomes a joint pdf.

For the to be true, the following criteria needs to be satisfied

i.e.

Where,

.

Now,

Hence;

.

(b) To show, X,Y are independent, we need to show that, , such that, .

chose, . Then, and hence proved.

(c)

(d) marginal distribution of y:

where,

(e)