Derive the density function of the random variable z y1y2

Question

Question 1: The management of a fast-food outlet is interested in the joint behavior of the random variables Y₁ = total time between a customer's arrival at the store and departure from the service window; and Y₂ the time a customer waits in line before reaching the service window. Because Y₁ includes the time a customer waits in line, we must have Y₁ ≥ Y₂. The bivariate density function for Y₁ and Y₂ is given by:

                     e^-y₁, 0 ≤ y2 ≤ y1 < ∞

f(Y₁, Y₂)= {

                     0,    Otherwise

with time measured in minutes.

a. Find P(Y₁ < 2,  Y₂ > 1).

b. Find P(Y₁ ≥ 2Y₂)

c. Find P(Y₁ ≥ Y₂ ≤ 1)

Question 2:

Consider two random variables Y₁ and Y₂ with the density function

                    2(1- y₁), 0 ≤ y₁ ≤ 1, 0 ≤ y₂ ≤ 1

f(Y₁, Y₂) =  {

                       0,                 otherwise.

Derive the density function of the random variable Z = Y₁Y₂

Question 3:

Let X₁, X₂, ...X_n be n random variables which are pairwise independent. Then

V(Σ_i=1ⁿX_i) = Σ_i=1ⁿV(X_i).