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 fXjX+Y 3.
(b) Find E[X j Y = y] for 1 y 3 and E[Y j X = x] for 1 x 4.
(c) Find cov(X; Y ).
2. Joint Density for Exponential Distribution
(a) If X Exponential( ) and Y Exponential( ), X and Y independent, compute P(X <Y).
(b) If Xk, 1 k n are independent and exponentially distributed with parameters
1; : : : ; n, show that min1 k n Xk Exponential(Pn j).
j=1
(c) Deduce that
P
(Xi =
min Xk) =
i
n
j
1 k n
Pj=1
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( ).
(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?
1
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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