EECS 126: Probability and Random Processes Problem Set 3

Description

1. Graphical Density

Figure 1 shows the joint density f_X;Y of the random variables X and Y .

Figure 1: Joint density of X and Y .

1. Find A and sketch f_X , f_Y , and f_{XjX+Y 3}.

2. Find E[X j Y = y] for 1 y 3 and E[Y j X = x] for 1 x 4.

3. Find cov(X; Y ).

2. Joint Density for Exponential Distribution

1. If X Exponential( ) and Y Exponential( ), X and Y independent, compute P(X <Y).

2. If X_k, 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( ).

1. Show that the number of packets routed to A is Poisson distributed. With what parameter?

2. Are the number of packets routed to A and to B independent?

4. Gaussian Densities

1. 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.

2. 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.

1. Find the transition probabilities of the Markov chain.

2. Find its stationary distribution.

Hint: You can guess the stationary distribution before computing it.

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