In this problem, we will consider a simple model for the time (rounded up to the nearest second) needed for a server to complete simultaneous jobs. Let X denote the current number of jobs running on the server. The marginal PMF of X is P_X(x) = {5/8 x = 1 1/4 x = 2 1/8 x = 3 0 otherwise. The service time Y is a Geometric random variable whose parameter depends on the number of jobs. Specifically, if X = x, then Y is Geometrical (1-x/8). a) Write down the conditional PMF P_Y|X (y/x). b) Write down the joint PMF P_X, Y (x, y). c) Determine the marginal PMF PY(y). d) Let B be the event that Y is less than or equal to 2. Calculate P[B]. e) Determine the conditional PMF of Y given B, PY|B(y). f) Calculate the conditional expected value of Y given B, E[Y|B]. g) Calculate the conditional variance of Y given B, Var [Y|B]. h) Calculate the conditional expected value of Y given X = x, E[Y|X = x].