EE132B-HW Set #1 UCLA 2014 Fall Prof. Izhak Rubin

Problem 1

Let X denote a geometric random variable with parameter 1p (0, 1) such thatP (X = n) = p(1 p)n, for n = 0, 1, . . . .

(1) Calculate the mean directly.

(2) Calculate the variance directly.

(3) Calculate the moment generating function (Z-transform).

(4) Using the moment generating function, derive the mean and the variance.

Problem 2

Let X denote an exponential random variable with parameter [0,). Theprobability density function for X is given by fX(x) = e

x, for x > 0.

(1) Calculate the mean directly.

(2) Calculate the variance directly.

(3) Calculate the moment generating function (Laplace transform).

(4) Using the moment generating function, derive the mean and the variance.

Problem 3

Show that the sum of two independent Poisson random variables has a Poissondistribution. Let X and Y denote two Poisson random variables with parameterX and Y , respectively. (Hint: Assume that the random variables X and Y areindependent. Set Z = X + Y . Prove Z has a Poisson distribution and determine itsparameter.)

Problem 4

Suppose that the number of customers entering a department store in a day is arandom variable with mean of 50 customers/day. Suppose that the amounts of moneyspent by each one of these customers are statistically independent random variableswith mean $8 (per customer). Also assume that the amount of money spent by eachcustomer is independent of the number of customers to enter the store. Calculatethe expected amount of money spent by the customers that enter the store during asingle day.

1

EE132B-HW Set #1 UCLA 2014 Fall Prof. Izhak Rubin

Problem 5

Define X and Y to be two discrete random variables whose joint probability massfunction is given as follows:

P (X = m,Y = n) =e74m3nm

m!(nm)!,

for m = 0, 1, . . . , n and n = 0, 1, 2, . . . , while P (X = m,Y = n) = 0 for other valuesof m,n. Calculate the marginal probability mass functions for the random variablesX and Y . Check whether X and Y are statistically independent random variables.

Problem 6

Suppose users share a 1 Mbps link. Also suppose that the traffic generationprocess of each user alternates (independently of the other users) between periods ofactivity (active modes), when the user generates data at a constant rate of 100 Kbpsand periods of inactivity (inactive mode) when the user generates no data. Supposefurther that the user is active (independent of other users) only 10 percent of thetime.

(1) When circuit switching is used to allocate resources on the shared link, howmany users can be supported?

(2) Suppose now that users are supported by using accordance with a packet switch-ing method, so that the communications channel is shared on a statistical mul-tiplexing manner (rather than being shared by using the synchronous (fixed)multiplexing techniques employed by the circuit switching system). Assumenow that the communications channel is used to support a number of usersthat is 4 times larger than those supported under the circuit switching oper-ation. What is now the probability that more than 10 simultaneously activeusers cannot be accommodated at a given point in time? Equivalently, what isthe probability that 11 or more users are simultaneously in active mode at agiven time?

2