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20065ee131A_1_EE131AF06HW5

20065ee131A_1_EE131AF06HW5 - Assignment 5 EE 131A Due...

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Assignment 5, EE 131A Due: Monday November 20, 2006 Problem 1. (a) First prove that the memoryless property of a random variable X P [ X > k + j v v X > j ] = P [ X > k ] , k, j > 0 , is equivalent to the following relation: P [ X > k + j ] = P [ X > k ] P [ X > j ] . Prove that the geometric random variable satisfies this property. (b) Suppose that X is a discrete random variable that satisfies the memoryless property. Show that this implies that X is a geometric random variable. ( Hint: Let p = P [ X = 1] and q = 1 - p . By induction, show that P [ X k ] = 1 - q k , and consequently show that P [ X = k ] = q k - 1 p .) Problem 2. The life of a certain type of automobile tire is normally distributed with mean m = 34 , 000 miles and standard deviation σ = 4000 miles. (a) What is the probability that such a tire lasts over 40,000 miles? (b) What is the probability that it lasts between 30,000 and 35,000 miles? (c) Given that it has survived 30,000 miles, what is the conditional probability that it survives

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20065ee131A_1_EE131AF06HW5 - Assignment 5 EE 131A Due...

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