EE 5375/7375 Random Processes due September 19, 2008

Homework #2

Problem 1. textbook problem 3.12Let U be a uniform random variable in the interval [-1,1]. Find the probabilities (a) P (U > 0)(b) P (U < 5) (c) P (|U | < 1/3) (d) P (1/3 < U < 1/2) (e) P (|U | ≥ 3/4).

Problem 2. textbook problem 3.15A random variable Y has the PDF

FY (y) ={

0 if y < 11− y−n if y ≥ 1

where n is a positive integer. (a) Plot the PDF of Y . (b) Find the probability P (k < Y ≤k + 1) for a positive integer k.

Problem 3. textbook problem 3.16A continuous random variable X has the PDF

FX(x) =

0 if x ≤ −π/2c(1 + sin(x)) if −π/2 < x ≤ π/21 if π/2 ≤ x

(a) Find c. (b) Plot FX(x).

Problem 4. textbook problem 3.18Let X be an exponential random variable with parameter λ. (a) For d > 0 and a positiveinteger k, find the probabilities P (X ≤ d), P (kd ≤ X ≤ (k + 1)d), and P (X > kd). (b)Segment the positive real line into 5 equiprobable disjoint intervals.

Problem 5. textbook problem 3.19A random variable X has the pdf

fX(x) ={cx(1− x) if 0 ≤ x ≤ 10 otherwise

(a) Find c. (b) Find P (1/2 ≤ X ≤ 3/4). (c) Find FX(x).

Problem 6. textbook problem 3.21A random variable X has the pdf shown in Fig. P3.2. (a) Find fX(x). (b) Find the PDF ofX. (c) Find b such that P (|X| < b) = 1/2.

Problem 7. textbook problem 3.27Let X be the exponential random variable with the pdf

fX(x) = λe−λx

1

fx(x)

-a 0x

Fig. P3.2 for Problem 6

a

!

Note: "(x) =1

2#$%

x

& e$t 2 / 2

dt =1

2+ erf (x)

c

for x ≥ 0. (a) Find and plot FX(x|X > t). How does FX(x|X > t) differ from FX(x)? (b)Find and plot fX(x|X > t). (c) Show that P (X > t + x|X > t) = P (X > x). Explain whythis is called the memoryless property.

Problem 8. textbook problem 3.34Let N be a geometric random variable with PMF

PN(x) = p(1− p)x

for x = 0, 1, . . .. (a) Find P (N > k). (b) Find the PDF ofN . (c) Find P (N is an even number).(d) Find P (N = k|N ≤ m).

Problem 9. textbook problem 3.35Prove the memoryless property of the geometric random variable:

P (M ≥ k + j|M > j) = P (M ≥ k)

for all j, k > 0. In what sense is M memoryless?

Problem 10. textbook problem 3.42The rth percentile, Pr, of a random variable X is defined by P (X ≤ Pr) = r/100. (a) Findthe 90%, 95%, and 99% percentiles of the exponential random variable with parameter λ.(b) Repeat part (a) for the Gaussian random variable with parameters m = 0 and σ. (Note:the table below may be handy.)

2

fx(x)

-a 0x

Fig. P3.2 for Problem 6

a

!

Note: "(x) =1

2#$%

x

& e$t 2 / 2

dt =1

2+ erf (x)

Problem 11. textbook problem 3.45A communication channel accepts an arbitrary voltage v and outputs a voltage Y = v + Nwhere N is a Gaussian random variable with mean 0 and variance σ2 = 1. Suppose thatthe channel is used to transmit binary information in this way: an input of -1 causes atransmission of 0, and an input of +1 causes a transmission of 1. The receiver decides a 0was sent if the voltage is negative and a 1 otherwise. Find the probability of the receivermaking an error if a 0 was sent; and if a 1 was sent.

Problem 12. textbook problem 3.47Messages arrive at a center at an average rate of one message per second. Let X be the timefor the arrival of 5 messages, i.e., time of the 5th message. Find the probability that X < 6;and that X > 8. Assume that message interarrival times are exponential random variables.

Problem 13. exponentialThe PDF of a random variable X is given by FX(x) = 1− e−x for x ≥ 0. The probability ofthe event {X < 1 or X > 2} is (a) 0.914 (b) 0.767 (c) 0.632 (d) 0.135 (e) 0.041.

Problem 14. normal

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A radar scans the skies for UFOs (unidentified flying objects). Let M be the event that aUFO is present and M c the complementary event that a UFO is absent. Let

fX|M(x|M) =1

2πe−(x−r)2/2

be the conditional pdf of the radar return signal X when a UFO is actually there and let

fX|Mc(x|M c) =1

2πe−x

2/2

be the conditional pdf of the radar return signal X when there is no UFO. To be specific, letr = 1 and let the alert level be xA = 0.5. Let A denote the event of an alert, i.e., {X > xA}.Compute P (A|M), P (Ac|M), P (A|M c), and P (Ac|M c). You might find the tables of thenormal distribution function below useful.

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