20065ee131A_1_EE131AF06HW6

# 20065ee131A_1_EE131AF06HW6 - Assignment 6 EE 131A Due...

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Assignment 6, EE 131A Due: Monday November 27, 2006 Problem 1. If a random variable X is such that E [( X - 1) 2 ] = 10 and E [( X - 2) 2 ] = 6 , find E [ X ] and Var [ X ] . Problem 2. A random variable X has the density function f X ( x ) = 1 2 e -| x | , -∞ < x < . (a) Find the distribution function of X . (b) Find the mean and variance of X ( Hint : use the formula i x 2 e a x dx = e a x ( a 2 x 2 - 2 a x + 2) /a 3 .) (c) Find P [ | X | > 2] . (d) Use Chebyshev’s inequality to obtain an upper bound on P [ | X | > 2] and compare with result in (c). Problem 3. Suppose that X is a random variable with mean and variance both equal to 20. Find a lower bound for P [0 X 40] . Problem 4. Suppose that X is non-negative integer-valued discrete random variable. Show that E [ X ] = s k =0 P [ X > k ] . Hint: First rewrite the right-hand summation as k =0 j = k +1 P [ X = j ] , and then find out for every n 0 the term “ P [ X = n ] ” how many times appears in the right-hand side. Problem 5.

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20065ee131A_1_EE131AF06HW6 - Assignment 6 EE 131A Due...

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