20105ee132B_1_hw1

20105ee132B_1_hw1 - UCLA Electrical Engineering Department...

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UCLA Electrical Engineering Department EE132B HW Set #1 Professor Izhak Rubin Problem 1 Let X denote a geometric random variable with parameter ( ) 10 p −∈ , 1 ) such that () ( 1 n P Xn p p == − for n = 0, 1,… 1) Calculate the mean directly. 2) Calculate the variance directly. 3) Calculate the moment generating function (Z-transform). 4) Using the moment generating function, derive the mean and the variance. Problem 2 Let X denote an exponential random variable with parameter ( ) 0, λ . The probability density function for X is given by x X f xe = for x > 0. 1) Calculate the mean directly. 2) Calculate the variance directly. 3) Calculate the moment generating function (Laplace transform). 4) Using the moment generating function, derive the mean and the variance. Problem 3 Show that the sum of two independent Poisson random variables has a Poisson distribution. Let X and Y denote two Poisson random variables with parameter X and Y , respectively. Assume that the random variables X and Y are independent. Set Z = X + Y . Prove Z
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20105ee132B_1_hw1 - UCLA Electrical Engineering Department...

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