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A probability quiz from the university of illinois at urbana-champaign's ece 434: random processes course, held in fall 2005. The quiz covers topics such as finding cumulative distribution functions, calculating expected values, and determining independence and conditional densities of random variables.
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Fall 2005 Probability Quiz
Monday, September 12, 2005
Name:
Score:
Total: (26 pts.)
Problem 1 (6 points) Let X have the pdf fX (x) =
{ (^) sin(x) 2 x^ ∈^ [0, π] 0 else (a) Find the cumulative distribution function FX. In particular, what is FX (2π)?
FX (2π) =
(b) Compute E[sin(X)].
Problem 3 (8 points) Suppose a N (0, 9) random variable X is passed through the quantizer function f shown. The output is Y = f (X).
! 1 ! 2 1 2
1
!0.
! 1
Y=f(x)
x
(a) Express the pmf of Y in terms of the Q function. (Check your answer. Make sure your answer is positive for the right values of y.)
(b) Express the variance of Y in terms of the Q function. Give as simple an answer as possible.