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(Pn

j=1 λj ).

(c) Deduce that

P(Xi = min

1≤k≤n

Xk) = P

λi

n

j=1 λj

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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