Question

1) Calls arrive at a help desk at an overall rate of per hour, where is a random variable with the p.d.f.

Let Xi be the number of calls that arrive during the ith hour for i=1,…,n. Supposed that are conditionally independent given = , each with the following conditional p.d.f.

a. Find the conditional mean and the variance given by .
b. Suppose that the manager wants to predict using a function after observing . Assume that she is subject to a loss of [ ]2. Find the value of the prediction that minimizes the conditional expected value of the loss and the overall mean square error, E[ ]2.
2) Suppose that X is a discrete random variable and Y is a continuous random variable. The conditional p.f. of X given Y is . The conditional p.f. of Y given X is . From the covariance of X and Y, Cov(X,Y).
3)
a. Show that Cov(X,Y) = E[Cov(X,Y|Z)]+Cov(E(X|Z),E(Y|Z)).
b. Supposed that be i.i.d. exponential random variables with parameter 1 and take N to be independent Poisson random variable with parameter that is independent of the Xi’s. Define compound Poisson random variable . Find the mean and the variance of S.
c. Supposed is a sequence of i.i.d N(0,1), and let N be Poisson random variable with parameter 2 that is independent of the sequence Xi, . We will define compound r.v. S as (b). Find the mean of S^2, E(S^2), Var(S2).
d. Supposed , are mutually independent r.v. such that Xk’s have a Bernoulli(1/2) distribution and the Yk’s have a Poisson(1) distribution. Let and .
i. Find the distribution of S and T
ii. Find P(S=T).
4)
a. The joint pdf of two r.v (X,Y) is given by f(x,y) = 6y, 0<1. Compute var(X-Y), and E(X-Y).
b. Joint p.d.f of (X,Y) is given by f(x,y) = K(x+y), 0<1. Find K such that this is a valid p.f.d. Compute var(X-Y), E(X-Y).
c. Supposed, we change the joint p.d.f. of (b) to f(x,y) = K*xy, 0<2y<4. Find value of K. Derive the joint distribution of U=X/Y and V=X. Verify that the joint pdf integrates 1. Are U and V independent? Explain your reason!
5)
a. A r.v. T is selected that is uniformly distributed over the interval . Then a second r.v. U is chosen, uniformly distributed on the interval . What is the probability that U exceeds ?
b. Let U be uniformly distributed over the interval where L follows the gamma density What is the joint density functions of U and V=L-U? What is the marginal density function of V=L-U?

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