Question 4 20 pts 4.a (20 points} Consider a protocol that works much like the selective repeat protocol (SRP) discussed in Set 8 of the class notes, but with more protection in retransmissions. The protocol is such that the original packet transmission succeeds with probability p and any retransmission of a packet succeeds with probability r > ,0, because of added coding protection used in retransmissions. As a result, the sender spends one transmission for the first transmission of a packet and with probability pthe receiver gets it. With probability 1 pthe sender spends one transmission and goes to the retransmit state. In that state, the sender spends one transmission for each retransmission of the packet and with probability rthe receiver gets it. Otherwise. with probability 1 rthe sender goes back to the same retransmit state and tries again subject to the same probabilities. Let Xbe the random variable denoting the number of transmissions (original and retransmissions) needed to receive a packet correctly at the receiver. Compute EX} when p = 0.2 and g: 0.8. Use the method we covered related to the law of total probability. D Question 6 15 pts (Extra 15 Points) The joint PDF of random variables X and Y is given by fx, x (x, y) = J XBe (ArtBy) ifx 20; y20 otherwise Find the PDF of the function W = Y / X and then choose the correct choice for fw (w) when w > 0. HINT: Find the CDF of W first! a). AB (X + Bio) 2 b). 1 (X + Bw) 2 c). ABw (x + Bu) 2 d). 1 AB (X + Bw) Answer O a O d O b OCQuestion 3 20 pts 3.a (20 points) The joint pdf of a bivariate random variable (X, Y) is given by ke Ve by fx, y (x, y)= y if0 5 | Y = 2). You must obtain f (X| Y) first and submit submit it as part of your work in the attached pdf file. a). e 5 2 bl. e-5 c). 1 - e-5 d). e ? Answer O a O d Ob O cQuestion 5 20 pts S.b. (20 points] A storage system can be in idle state {0). writing state (1}, or reading state (2}. The arrival of requests to read or write files from the system is such that, on steady state. the probabilities that the system is in any one of such states is given by the homogeneous Markov chain shown in the figure below, which indicates the transition probabilities. Compute the state probability that the system is in writing state. A as war 0 11m 0 21mg 0 25M? o 5:12 D Question 1 20 pts 1.b. (20 points) Random variables Xand Yhave the joint PMF Px,y (x, y) = 10 Sexty| ifx=-2,0,2; y = -1,0,1 otherwise Compute the value of c and then determine the following probability: P [Y > X ] Answer O 1/4 O 4/7 O 1/2 O 1/3D Question 2 20 pts 2.a (20 points) The joint pdf of a bivariate random variable (X, Y) is given by (Ity) fx,x (x, y) = 3 ifox