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(elec201)[2008](f)final~PPSpider^_10290.pdf
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ELEC210 C Probability and Random Processes in Engineering
Final Examination
December 20, 2008
Duration for the examination is 3 hours
There are altogether 5 questions
Total score is 105 points
Name:
Student Number:
Question
Your Mark
Mark of Question
Q1
20
Q2
20
Q3
25
Q4
20
Q5
20
Total
105
Problem 1. (20 points)
The pdf of a random variable X is shown below.
a)
(2 pts) Let ()XFxbe the cdf of the random variable X. Determine the value of (1)XF.
b)
(3 pts) Because the pdf of X is symmetrical around x=0, E[X]=0. Hence, 2. Show that .
c)
(3 pts) Let. Sketch ()Yfy, the pdf of Y. Carefully label all important dimensions in your sketch.
d)
(2 pts) Let ()g. be a clipper such that .
Let
. Sketch
, the pdf of Z. Carefully label all important dimensions.
slope = 0.5
slope = -0.5
x
e)
(2 pts) Sketch ()ZFz, the cdf of Z. Carefully label all important dimensions.
f)
(2 pts) Let W be a Bernoulli random variable that has probability of 0.6 to be 0 and probability of 0.4 to be 1. Assume that W is the input to a communication channel and X is an additive random noise such that the channel output is V = W + X. We assume that W and X are independent. Leveraging the result in part (b) and using the provided tables if necessary, find the variance of V.
g)
(3 pts) Find[0]PV<, the probability that the channel output is less than 0.
h)
(3 pts) Recall that Bayes Rule states that . Find], the conditional probability that W =1 given that V is less than 0.
Problem 2. (20 points)
a)
(2 pts) Let ()h.be a function that is 0.5 times the absolute value of its argument. That is, |. The function ()h. can be represented by the function below:
Let W = h(X). W is also a random variable. Let A be the event in W that . Taking reference to the diagram above, sketch and show the equivalent event(s) in X that gives rise to A.
b)
(6 pts) The pdf of X is given as below:
Recall that for W being a function of X,
, where
s are all the solution to . Find
, the pdf of W = h(X). Provide the mathematical expression as well as a sketch. Label all important dimensions.
11
-1 2
2/3 t
0
T=1
For the parts below, let be zero-mean, jointly Gaussian random variables with covariance matrix