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Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Problem Set 3
1. Graphical Density
Figure 1 shows the joint density fX,Y of the random variables X and Y .

Figure 1: Joint density of X and Y .
(a) Find A and sketch fX, fY , and fX|X+Y ≤3.
(b) Find E[X | Y = y] for 1 ≤ y ≤ 3 and E[Y | X = x] for 1 ≤ x ≤ 4.
(c) Find cov(X,Y ).
2. Joint Density for Exponential Distribution
(a) If X ∼ Exponential(λ) and Y ∼ Exponential(µ), X and Y independent, compute P(X < Y ).
(b) If Xk, 1 ≤ k ≤ n are independent and exponentially distributed with parameters
λ1,…,λn, show that min1≤k≤n Xk ∼ Exponential( (c) Deduce that

3. Packet Routing
Packets arriving at a switch are routed to either destination A (with probability p) or destination B (with probability 1−p). The destination of each packet is chosen independently of each other. In the time interval [0,1], the number of arriving packets is Poisson(λ).
(a) Show that the number of packets routed to A is Poisson distributed. With what parameter?
(b) Are the number of packets routed to A and to B independent?
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4. Gaussian Densities
(a) Let X1 ∼ N(0,1), X2 ∼ N(0,1), where X1 and X2 are independent. Convolve the densities of X1 and X2 to show that X1+X2 ∼ N(0,2). Remark. Note that this property is similar to the one shared by independent Poisson random variables.
(b) Let X ∼ N(0,1). Compute E[Xn] for all integers n ≥ 1.
5. Moving Books Arround
You have N books on your shelf, labelled 1,2,…,N. You pick a book j with probability 1/N. Then you place it on the left of all others on the shelf. You repeat the process, independently. Construct a Markov chain which takes values in the set of all N! permutations of the books.
(a) Find the transition probabilities of the Markov chain.
(b) Find its stationary distribution.
Hint: You can guess the stationary distribution before computing it.
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