1. (a) W = log (1). Here log is log base e. i. Find the range of W. ii. Find the density of W. The joint distribution of two random variable X and Y is (b) Part (a) and (b) of this question are independent. Suppose a random variable U follows Uniform(0,1) distribution. Define fxy(x, y) = 8 xy 1(0

3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.

Answer part a of the following question. Show work. Let Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X ⊥ Y and model time to death as X ∼ Exp(rate = θ) and time to transplant as Y ∼ Exp(rate = µ). Where Exp(rate = λ) denotes the exponential distribution with density f(z | λ) = λe−λz for z > 0 and 0 elsewhere - with λ > 0. a.) Compute the probability that a patient receives transplant before they pass away (die) by integrating over the joint density fxy.

Let X and Y be independent random variables with the distributions X ∼ Geometric(p = 1/3)and Y ∼ Geometric(p = 2/3). Find Pr[X = Y].