Problem #1: Let X be a Poisson random variable with parameter 3, and Y a geometric random variable with parameter 11⁄2 3. Suppose that X and Y are independent, and W = X + Y . Find P{W X+Y. = 3}. Problem #1: Round your answer to 4 decimals.

Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(X)=0 Var(X) = 39 Var(Y) = 19 Var(Z) =24 E(Y) =-4 E(Z)=6 Compute the values of the expressions below. E(1+5z) = -Y-4Z E 2 Var(4–3Y) = I E(sz2) = I

Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(Y)=-7 Var (X) = 20 Var (Y)=6 Var (z) = 23 E (X) = 5 E(Z) = 3 Compute the values of the expressions below. E (-2X – 4) = -3Z + 2X E 3 Var (27 + 3) - O =(-sx") - O

7) Suppose that X and Y are two independent measurements. We can treat these measurements as random variables, where X has the mean of 1 and standard deviation of 2, and Y has the mean of 2 and the standard deviation of 3. For the transformations below, find the mean, variance, and the standard deviation. a) X+Y b) X-Y c) 2X+Y d) X-2Y+1

Question 3. Let X1,, Xn be a random sample from a distribution with the pdf given by f11x(x) = exp(-a) if x ≥ A, otherwise f(x) = 0, where> 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.

(a) Recall that for a random variable X and constants a and b, E(aX+b) = aE(X) +b. Prove that Var(aX + b) = = a2Var(X). (b) Mike agrees to donate 50 dollars to charity, plus 25 dollars for every goal scored in a particular World Cup soccer game. Thus Mike's donation is a random variable Y = 25X + 50 where X is defined in problem 1. Find the expected value and the variance of Y.

Problem #2: If two loads are applied to a cantilever beam as shown in the figure below, the bending moment at 0 due to the loads is a1X1 + a2X2. Problem =2(a) Problem =2(b) X1 Problem #2(c): a X2 9 Suppose that X1 and X2 are independent random variables with means 2 and 5 kips, respectively, and standard deviations 0.7 and 1.5 kips. respectively. Suppose that a1 = 3 ft and a2 = 10 ft. (a) Find the expected value of the bending moment. (b) Find the standard deviation of the bending moment. (c) If X1 and X2 are normally distributed, what is the probability that the bending moment will exceed 82 kip-ft? answer correct to 4 decimals answer correct to 4 decimals