ECE 3800: EXAM #1 Spring 2016
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   7
(a)
8
Pr b   8
10
Pr c   7
8
Pr d   6
10
Pr e   5
10
If switch b becomes stuck in the open position, new Pr b   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 a  b switch path is transmitting (both a and b closed)?
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ECE 3800 Exam#1
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2. [40 pts]
Consider an experiment where two four sided triangular dice are rolled with face
values of 1, 2, 3, and 4. The sample space consists of the sum of the two triangular die after
being rolled, Z=X+Y. Note they are “fair” die so the four sides have equal probability.
(a)
What is the probability mass function (pmf) of the numerical result, Z?
(b)
Compute the probability PrZ  4 .
(c)
Given that the second die is a 2 (Y=2), what is the probability that PrZ  4 which is
also defined as the conditional probability Pr Z  4 | Y  2 ?
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ECE 3800 Exam#1
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3. [50 pts]
A widget has a constant failure rate that can be describing using the exponential
density function:
x0
x0
0,
0,


and FX  x   
f X x    1
 1 
 1 
   exp    x ,0  x  
1  exp    x ,0  x  






Because it is worried that this is not sufficient for an important application, we are going to use
two widgets, where one takes over for the other once the first one fails. The resulting “expected”
lifetime of the product becomes Z=X1+X2. Instead of just X. Assume the Xi are independent and
identically distributed (IID).
Exponential hint
Integral hint

x  expax   dx  expax  
m
m
  1r 
r 0
m! x m  r
m  r !a r 1
 u  dv  u  v   v  du
(a)
Determine explicit formula for the new probability density function, f Z  z  .
(b)
Find the explicit formula for the cumulative distribution function, FZ z  .
(c)
For one original widget, calculate Pr  X  , it survives longer than lambda.
(d)
For the redundant system, calculate Pr  Z  and Pr2    Z  , the redundant system
survives longer than lambda and 2xlambda.
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ECE 3800 Exam#1
4. [20 pts]
(a)
(b)
Spring 2016
Consider the following probability density function:
x0
0,

f X x    1
 1 
   exp    x ,0  x  



Assume that Y is related to X as Y  2  X  2 . Now, derive the probability density
function for f Y  y  .
Calculate the probability PrY  0 .
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5. [30 pts])
Spring 2016
A Gaussian random voltage has a mean value of 2 volts (  X  2 ) and a
standard deviation of 2 volts (  X  2,  X2  4 ). (See page 8 for a standard normal CDF table)
(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 -2 volt but
less than or equal to 6 volts.
(c)
What is the probability that the observed value of the voltage is greater than 3 volts?
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