Question

1- The future lifetimes (in months) X and Y of two components of a machine have the following joint pdf

f(x,y)= (6/125,000) (50−x−y), if 0<x<50−y<50

0, elsewhere
Calculate the probability that both components are still functioning 20 months from now.

2- Let X and Y be two continuous random variables with joint pdf

f(x,y)= (8/3)xy, if0<x<1,x<y<2x

0, elsewhere

(i) Find the marginal pdf f1(x) of X. Specify its support.
(ii) Find the marginal pdf f2(y) of Y . Specify its support.
(iii) Are X and Y independent? Explain your answer.
(iv) Use your answers to (i) and (ii) to compute E(X), V ar(X), E(Y ) and V ar(Y ). (v) Compute the covariance, cov(X, Y ), of X and Y .

(vi) What is the correlation ρX,Y of X and Y
3- The joint density for the length of life of two different types of components operating

in a system is given by

f(x,y)= (1/8)xe*−(x+y)/2, if x > 0,y > 0

0, elsewhere

show that X and Y are independent

Hint: Show, by a simple inspection of the formula, that f(x,y) can be written as the product of two know univariate pdf’ in x and y, respectively.

Answer 1