EE132B-HW Set #2 UCLA 2014 Fall Prof. Izhak Rubin

Problem 1For the Gaussian distribution with mean and variance 2, find the moment

generating function. Using the moment generating function, calculate the mean andthe variance. (Hint: The probability density function for a Gaussian random variableX with mean and variance 2 is given by

fX(x) =12pi2

exp((x )

2

22

).

Problem 2Consider the following probability density function:

fX(x) =e|x|

2 , x (,),

where > 0.

(a) Calculate the mean directly.

(b) Calculate the variance directly.

(c) Find the moment generating function.

(d) Find the mean and the variance from the moment generating function.

Problem 3A coin is flipped until heads occur twice. Define two random variables X and Y

to be the trial numbers at which the first and the second heads are observed. Assumethat at any trial, the probability that a head occurs is p (0, 1).(a) Show that the joint probability is given by

P (X = m,Y = n) ={p2qn2 ,m = 1, 2, . . . ; n = 2, 3, . . . ,0 , otherwise.

(1)

(b) Calculate the marginal probability mass function for X and Y .

(c) Calculate the conditional probability that X = m, given Y = n (i.e., P (X =m | Y = n)).

1

EE132B-HW Set #2 UCLA 2014 Fall Prof. Izhak Rubin

Problem 4Two DTEs are communicating via modems and RS-232C interfaces over a half

duplex link. Data transmission is one directional with each message of length M .After a message is transmitted, the sender waits for a CTS message of length na beforesending another message. Assume error-free operation and the following parametervalues:

R (line bit rate) = 9600 bps M (message length) = 1000 bits (all data bits) nh (overhead in each message packet) = 24 bits na (acknowledgement length) = 24 bits tp (propagation time in one direction) = 1 msecMinimum transition time between states (1,3) and (1,1) (CTS delay) is equal to

10 msec at message transmitting end, and 20 msec at message receiving end (CTStransmitting end). All other delays during message transmission and reception areassumed to be negligible. For the stated conditions, determine the effective data ratein bps.

2