Let X1 exp1 and X2 exp1 be independent and identicallydistr

Let X1~ exp(1) and X2 ~ exp(1) be independent and identically-distributed exponential random variables with rate 1.

Let: Y = X1 + X2 , Z = X1 X2

(a) What is the cdf of X1?

(b) What is the joint pdf of (X1, X2)?

(c) What is the joint pdf of (Y, Z)?

(d) What is the marginal pdf of Z?

Solution

a. X1~ exp(1), then cdf of X₁ is F(x) = e^-x1+1

b. X₁~ exp(1), then pdf is f(x)=e^-x1

X₂~ exp(1), then pdf is f(x)=e^-x2

the joint pdf of (X1, X2)= e^-x1e^-x2

^c. Y=( e^-x1 +e^-x2)

Z=( e^-x1 - e^-x2)

the joint pdf of (Y, Z)= ( e^-x1 +e^-x2)( e^-x1 - e^-x2)

   ⁼ e^-2(x1+x2)

^d. marginal pdf of Z= x₁( e^-x1 - e^-x2)* x₂( e^-x1 - e^-x2)