20092ee132B_1_hw2

20092ee132B_1_hw2 - U CLA Electrical Engineering Department...

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UCLA Electrical Engineering Department EE132B HW Set #2 Professor Izhak Rubin Problem 1 For the Gaussian distribution with mean μ and variance 2 σ , find the moment generating function. Using the moment generating function, calculate the mean and the variance. Hint: The probability density function for a Gaussian random variable x with mean μ and variance 2 σ is given by ( 29 ( 29 2 2 1 2 2 1 2 x X f x e μ σ πσ - - = . Problem 2 Consider the following probability density function: ( 29 2 x X e f x λ λ - = for ( 29 , x ∈ -∞ ∞ . 1) Find the mean directly. 2) Find the variance directly. 3) Find the moment generating function. 4) Find the mean and the variance from the moment generating function. Problem 3 A coin is flipped until heads occur twice. Define two random variables X and Y to be the trial numbers at which the first and the second heads are observed. Assume that at any trial, the probability that a head occurs is ( 29 0,1 p .
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