Question

Let X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter λ=3. Furthermore assume X and Y are independent.

The cumulative distribution of Z = X + Y is

a. P(Z≤a)=P(X+Y≤a) = _____ for 0 < a < 1
b. P(Z≤a)=P(X+Y≤a) = _____ for 1

Answer 1

The cumulative distribution of Z, which is defined as Z = X + Y, can be found using the properties of the uniform and exponential distributions and the fact that X and Y are independent.

Step-by-step explanation:

The cumulative distribution of Z, which is defined as Z = X + Y, can be found using the properties of the uniform and exponential distributions and the fact that X and Y are independent.

For 0 < a < 1, P(Z≤a) = P(X+Y≤a) = ∫∫(P(X+Y≤a|x)pX(x)pY(y)) dx dy = ∫∫(P(Y≤a-x)pX(x)pY(y)) dx dy = ∫(∫(P(Y≤a-x)pX(x)) dx) dy = ∫(P(Y≤a-x) dx) dy

For 1 < a, P(Z≤a) = 1, because Z cannot exceed the sum of the maximum values of X and Y.