Name

ID Number

Midterm – EE505

6 November 2014

Instructions:

• The test is open notes, but closed book.

• Show all work. Partial credit will be given for partial work; NO credit will be given for nowork.

• Be sure to state all assumptions made and check them when possible.

• There are five problems on seven pages, including the cover and a blank page is attached atthe end in case you need extra space.

Honor Code:This exam represents only my own work. I did not give or receive help on this exam.

Signature Date

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1. (20 points)The probability that the phone will be busy when you call ABC company is q. You keeptrying until you get through. The random variable X is the number of times that you haveto call before getting through.

(a) What is the probability that you need to make at least 10 calls?

(b) What is the probability that you need to make at least 5 more calls given that you’vealready made 5 calls?

(c) If the time that you wait between calls is exponential with mean 10 minutes, what is theexpected time it will take to get through from the time you start calling? Assume thatthe waiting time is independent of the call getting through, waiting times are i.i.d., andthe time making an unsuccessful call is always 1 minute.

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2. (20 points)LetX=X1, X2, X3, X4 be jointly Gaussian random variables with mean vector and covariancematrix given by:

mX =

2

6664

�1201

3

7775 ,KX =

2

6664

2 1 0 01 2 1 00 1 2 00 0 0 4

3

7775

(a) Which pairs of variables (if any) are independent? Which are orthogonal?

(b) Find the joint pdf of X2 and X3

(c) Find the pdf of Y = X1 �X2 +X4

(d) Find the conditional pdf of fY |X3(y|x3)

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3. (20 points)Survey workers sample independent people leaving a voting site. The probability that theyvote for candidate A is ↵.

(a) Given that N = 500 people truthfully respond to the survey, what is the expectednumber of people who say they voted for candidate A.

(b) The pollsters estimate ↵̂ = 0.55 as the sample mean of those 500 responses. Explain howyou would use the Central Limit Theorem to estimate the confidence that candidate 1actually won. (In other words, what is the probability that ↵̂ is more than 10% greaterthan the true ↵?) Give your answer in terms of the Q function.

(c) If the number of people leaving a polling site is Poisson with parameter �, and there are5 independent polling sites, and the probability that someone will talk to a pollster isp = 0.5, what is the probability of getting at least 500 responses? (Leave your answerin the form of a sum.)

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(d) As it turns out, people are not always truthful, and the probability that they tell thetruth depends on if they think they are in the majority. Let X be a random variableindicating how someone votes, and Y be the random variable indicating what they say,i.e. X = 1 indicates the voted for candidate A (or 0 for the other candidate) and Y = 1indicates that they say they voted for candidate A. Assume P (Y = 1|X = 1) = 0.9,P (Y = 0|X = 0) = 0.6, and P (X = 1) = ↵ = 0.5. What is the probability that they saythey vote for candidate 1? Find the correlation E[XY ].

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4. (20 points)Let X(t) = e

�Atu(t), where u(t) is the unit step function and A is a random variable with an

exponential distribution with parameter �.

(a) Find E[X(t)] and RX(t, s) for 0 < t < s.

(b) Find the sample space and the CDF for X(t) for t > 0.

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5. (20 points)For each random process below, determine whether Y (1) and Y (5) are uncorrelated.

(a) {Y (n)} is a discrete-time random process with constant mean mY = �1 and auto-

covariance CY (n, n+ k) = 23|k|.

(b) Y (n) =Pn

i=1X(i) where {X(n)} is i.i.d. Gaussian with zero mean and variance �

2.

(c) Given Q = q, the sequence {Y (n)} is a Bernoulli process with parameter q, and Q is arandom variable with a uniform (0,1) distribution, i.e. q is unknown and fixed for all nin a sequence.

(d) {Y (n)} is a discrete-time random process with constant mean mY = 1 and auto-correlation

RY (n,m) =

(2 n = m

1 n 6= m

7

Extra Work Space

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