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Unformatted text preview: EE 278 Handout #3 Statistical Signal Processing Monday, July 6, 2009 Homework Set #2 Due: Monday, July 13, 2009. 1. Random phase signal Let Y ( t ) = sin( ωt +Θ) be a sinusoidal signal with random phase Θ ∼ U [ − π,π ] . Find the pdf of the random variable Y ( t ) for fixed values of time t and radial frequency ω . Comment on the dependence of the pdf of Y ( t ) on time t . 2. Quantizer Let X ∼ Exp( λ ) be an exponential random variable with parameter λ . Let Y = ⌊ X ⌋ be the integer part of X ; i.e., if k ≤ X < k +1 then Y = k for k = 0 , 1 , 2 ,... . (a) Find the pmf of Y . (b) Define the quantization error Z = X − Y . Find the pdf of Z . 3. Gambling Alice enters a casino with one unit of capital. She looks at her watch to generate a random variable U ∼ U[0 , 1] , then bets the amount U on a fair coin flip. Her wealth X is thus given by X = braceleftBigg 1 + U with probability 0.5 1 − U with probability 0.5 Find the cdf of X . 4. Generating an exponential r.v. from a Gaussian Let X ∼ N (0 , 1). Find a function g ( x ) such that Y = g ( X ) is exponentially distributed with parameter λ , i.e., Y ∼ Exp( λ ). Express your answer in terms of the Q ( · ) function. 5. Joint cdf or not Consider the function F X,Y ( x,y ) = braceleftBigg 1 if x + y ≥ otherwise. Can F X,Y ( x,y ) be a joint cdf for a pair of random variables? Justify your answer. Hint: Check if the function satisfies the properties of a joint cdf given in the lecture notes. 6. Marginal and conditional pdf Consider random variables X and Y with joint pdf f X,Y ( x,y ) = braceleftBigg c if | x | + | y | ≤ 1 / √ 2 otherwise. where c > 0 is a constant. (a) Find c . (b) Find f X ( x ) and f X | Y ( x | y )....
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This note was uploaded on 10/26/2011 for the course ECE 180 taught by Professor Eggers during the Spring '11 term at Aarhus Universitet.
- Spring '11
- Signal Processing