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# hw2 - EE 278 Statistical Signal Processing Homework#2 Due...

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EE 278 September 30, 2009 Statistical Signal Processing Handout #2 Homework #2 Due: Wednesday October 7 1. Probabilities from cdf. The cdf of random variable X is given by F X ( x ) = braceleftBigg 1 3 + 2 3 ( x + 1) 2 - 1 x 0 0 x < - 1 a. Find the probabilities of the following events. A = { X > 1 3 } , B = {| X | ≥ 1 } , C = {| X - 1 3 | < 1 } , D = { X < 0 } . b. Does X have a pdf? Explain your answer. 2. Distance to nearest star. (Bonus) Let the random variable N be the number of stars in a region of space of volume V . Assume that N is a Poisson random variable with pmf p N ( n ) = e - ρV ( ρV ) n n ! , n = 0 , 1 , 2 , . . . , where ρ is the “density” of stars in space. We choose an arbitrary point in space and define the random variable X to be the distance from the chosen point to the nearest star. Find the pdf of X in terms ρ . 3. Additive Gaussian noise channel. A communication channel has a real-valued input signal X and an output Y = X + 2 Z , where Z ∼ N (0 , 0 . 09). Suppose that X = - 1 is sent. Use the attached table of the Q ( · ) function to find the probability of the event { Y > 0 } . 4. Lognormal pdf. Let X ∼ N (0 , σ 2 ). Find the pdf of Y = e X . 5. Random phase signal. (Bonus) Let Y ( t ) = sin( ωt + Θ) be a sinusoidal signal with random phase Θ U[ - π, π ] . Find the pdf of the random variable

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hw2 - EE 278 Statistical Signal Processing Homework#2 Due...

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