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Homework Assignment 1 - Homework Assignment 1(1 Let f(x =...

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Homework Assignment 1: (1) Let f(x) = a (e -2x – e -3x ) for x 0 and 0 elsewhere. Find the constant a so that f(x) satisfies the conditions of being a probability density function. Calculate Pr(X 1). (Ans a = 6; Pr(X 1) = 0.6936) (2) Let f(x) = 1.5x + .25, 0 x 1, zero elsewhere, be the probability density function of X. Find the distribution function of X and calculate Pr ( ¼ X ¾). (Ans F(x) = 0 for x < 0, .75x 2 + .25x for 0 x 1 and 1 for x > 1. Pr( ¼ X ¾) = 0.5) (3) You play a slot machine repeatedly. The probability of winning on a single play is 0.05, and successive plays are independent. Let X be the random variable for the number of unsuccessful attempts before the first win. Find F(x) for x = 0,1,2,… , and determine how many times you would need to play the slot machine in order to be sure that your probability of winning at least once is 99%. (Ans. F(x) = 1 - .95 x+1 , x = 0,1,2,… . You need 90 plays for the probability to be 99% that you win at least once.) (4) Consider the following experiment: Start by tossing a fair coin.
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