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Problem Set 3 Graphical Density

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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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