ECE 3800: EXAM #1 Fall 2016 - Homepages at WMUbazuinb/ECE3800SW/Exam1.pdfECE 3800 Exam#1 Fall 2016 ECE 3800, Probabilistic Methods of Signal and System Analysis Page 3 of 8 2. [40

ECE 3800: EXAM #1 Fall 2016

ECE 3800, Probabilistic Methods of Signal and System Analysis Page 1 of 8

1. [40 pts]. A communication network has three paths controlled by five switches as shown below. In order to communicate, one or more paths must be fully connected between X and Y. The probability that each switch is closed is:

108Pr a 8

6Pr b 105Pr c 8

5Pr d 108Pr e

a b

c

d e

X Y

(a) If switch c becomes stuck in the open position, new 100Pr c , what is the probability

that an electrical signal transmitted from node X will be received at node Y?

(b) For all switches working based on their probabilities, what is the probability that an electrical signal transmitted from node X will be received at node Y?

(c) For all switches working based on their probabilities, what is the probability that an electrical signal will not be able to flow from node X to node Y?

(d) If you know that a signal is being transmitted from node X to node Y, what is the conditional probability that switch c is closed?

ECE 3800 Exam#1 Fall 2016


1 continued)



2. [40 pts] Consider an experiment where there is (#1) one four sided triangular dice with face values of 1, 2, 3, and 4 and (#2) one six sided dice with face values 1, 2, 3, 4, 5, and 6. The sample space consists of the sum of the die after being rolled, Z=X+Y. Note they are “fair” die so the sides have equal probability.

(a) What is the probability mass function (pmf) of the numerical result?

(b) Compute the probability 5Pr Z .

(c) Given that the second die (#2) is a 3 (Y=3), what is the probability that 6Pr Z which

can also be defined as the conditional probability 3|6Pr YZ ?



3. [50 pts] Occasionally, a redundant system will have a high quality part as the primary component and a lesser quality or performance part as the spare. For two parts X (higher quality) and Y having exponential pdf failure rates defined below. We want the expected pdf failure rate for the redundant system Z=X+Y. Let,

xxaa

xxf X 0,exp

0,0 and

yybb

yyfY 0,exp

0,0

Note that for Y expected to fail before X, we can expect that b > a.

Exponential hint

m

rr

rmrm

arm

xmaxdxaxx

01!

!1expexp

(a) Determine explicit formula for the new probability density function, zfZ .

(b) Find the explicit formula for the distribution function, zFZ .

(c) For a single primary component X, calculate Xa 21Pr .

(d) For the redundant system Z, calculate Za 21Pr , Assume that ab 2 .



3 continued)



4. [30 pts] Consider the following probability density function:

x

xxxA

x

xf X

3,0

31,32

1,02

(a) Determine the value of A that makes this a proper probability density function.

(b) Assume that Y is related to X as 22

1 XY . Now, derive the probability density

function for yfY .

(c) Calculate the probability 1Pr Y .



5. [30 pts]) A Gaussian random voltage has a mean value of 6 volts ( 6X ) and a standard deviation of 5 volts ( 25,5 2 XX ).

(a) What is the probability that the observed value of the voltage is less than or equal to zero volts?

(b) What is the probability that an observed value of the voltage is greater than 1 volt but less than or equal to 13.5 volts.

(c) What is the probability that the observed value of the voltage is greater than 8 volts?



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ECE 3800: EXAM #1 Fall 2016 - Homepages at WMUbazuinb/ECE3800SW/Exam1.pdfECE 3800 Exam#1 Fall 2016 ECE 3800, Probabilistic Methods of Signal and System Analysis Page 3 of 8 2. [40