Digital Assignment - 1 ECE2005 – Probability, Theory and Random Process 1. Random variables X and Y are distributed according to the joint PDF (a) Evaluate the constant a. (b) Determine the marginal PDF fY (y). (c) Determine the conditional expectation of 1/X given that Y = 3/2. (d) Random variable Z is deﬁned by Z = Y − X. Determine the PDF fZ(z). 2. Let X and Y be two independent random variables. Their probability densities functions are shown below. 1 fX (x) = 34(1 − x2) fX (x) 0.8 0.6 0.4 0.2 0 −1.5 −1 −0.5 0 0.5 x 1 1.5 fY (y) 1 0.8 2 3 0.6 1 3 0.4 0.2 0 0 Let Z = X + Y . Determine fZ(z). 0.5 1 1.5 y 2 2.5 3 3. Random variables X and Y have the joint PDF shown below: y 2.0 fX,Y (x,y) = 0.1 1.0 1.0 -1.0 2.0 x -1.0 -2.0 (a) Find the conditional PDFs fY |X (y | x) and fX|Y (x | y), for various values of x and y, respectively. (b) Find E[X | Y = y], E[X], and var(X | Y = y). Use these to calculate var(X). (c) Find E[Y | X = x], E[Y ], and var(Y | X = x). Use these to calculate var(Y). 4. X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on values between 0 and 1. Their joint pdf is indicated below. 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x 1.2 1.4 1.6 1.8 2 (a) Are X and Y independent? Present a convincing argument for your answer. (b) Prepare neat, fully labelled plots for fX (x), fY |X (y | 0.5), and fX|Y (x | 0.5). (c) Let R = XY and let A be the event X < 0.5. Evaluate E[R | A]. (d) Let W = Y − X and determine the cumulative distribution function (CDF) of W . 5. A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF In essence, a speciﬁc coin produced by this machine will have a ﬁxed probability P = p of giving heads, but you do not know initially what that probability is. A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent Find the probability that the ﬁrst coin toss results in heads. (a) Given that the ﬁrst coin toss resulted in heads, ﬁnd the conditional PDF of P . (b) Given that the ﬁrst coin toss resulted in heads, ﬁnd the conditional probability of heads on the second toss. 6. The wombat club has N members, where N is a random variable with PMF pN (n) = pn−1 (1 − p) for n = 1, 2, 3, . . .. On the second Tuesday night of every month, the club holds a meeting. Each wombat member attends the meeting with probability q, independently of all the other members. If a wombat attends the meeting, then it brings an amount of money, M , which is a continuous random variable with PDF fM (m) = λe−λm for m ≥ 0. N , M , and whether each wombat member attends are all independent. Determine: (a) The expectation and variance of the number of wombats showing up to the meeting. (b) The expectation and variance for the total amount of money brought to the meeting.