ECE 3800: EXAM #1 Spring 2015
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:
Pr a   5
(a)
10
Pr b   5
10
Pr c   8
10
Pr d   8
10
Pr e   5
10
If switch d becomes stuck in the open position, new Pr d   0
, what is the probability
10
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 the d  e switch path is transmitting (both d and e closed)?
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#1
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1 continued)
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ECE 3800 Exam#1
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2. [40 pts]
Consider an experiment where two six sided die are rolled, but they are special
die with 3 ones, 2 twos and 1 three on the six faces. The sample space consists of the sum of the
two 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, Z?
(b)
Compute the mean of Z.
(c)
Compute the second moment of Z.
(d)
Compute the probability PrZ  3 .
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#1
3. [50 pts]
Spring 2015
Consider the following probability density function:
0,
f X x   
2  exp 2  x  1
x 1
1 x  
(a)
Find the explicit formula for the distribution function, FX  x  .
(b)
Find the mean of X, EX    X .
(c)
Find the 2nd moment, variance and standard deviation of X ( E X 2 ,  X2 and  X ).
(d)
Calculate the probability PrEX    X  X  EX    X .
(e)
3

Calculate the conditional probability Pr   X | X  2 .
2

 
Exponential hint
 x  expax   dx 
x
expax 
 ax  1
a2
m
m
 expax   dx  expax     1r 
m! x m  r
m  r !a r 1
expax  2 2
2
 x  expax   dx  3  a  x  2  a  x  2
r 0
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#1
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Exponential hint
 x  expax   dx 

expax 
 ax  1
a2
x m  expax   dx  expax  
m
  1r 
m! x m  r
m  r !a r 1
expax  2 2
2
 x  expax   dx  3  a  x  2  a  x  2
r 0
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#1
4. [20 pts]
Consider the following probability density function:
0,
1  x

f X x   
1  x
0
(a)
(b)
Spring 2015
x  1
1  x  0
0  x 1
1 x
Assume that Y is related to X as Y  abs X  .
Derive the probability density function for f Y  y  .
1

Calculate the probability Pr Y   .
2

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ECE 3800 Exam#1
Spring 2015
5. [30 pts])
(2-5.1 based) A Gaussian random voltage has a mean value of 1.5 volts
(  X  1.5 ) and a standard deviation of 3 volts (  X  3,  X2  9 ).
(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 - 1.5 volt but
less than or equal to 3.0 volts.
(c)
What is the probability that the observed value of the voltage is greater than 6 volts?
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