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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