pitman_fall_2005_final - Statistics 134 Fall 2005 Final...

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Statistics 134 Fall 2005 Final Exam Professor James Pitman * 1. A random variable X with values between - 1 and 1 has probability density function f ( x ) = cx 2 for x in that range, for some constant c . (a) Find c as a decimal. (b) Give a formula for the cumulative distribution function of X . (c) Find Var( X ) as a decimal. (d) Let Y = X 2 . Find the probability density function of Y . 2. A multiple choice test has 4 possible answers for each question, exactly one of which is right. The test has 20 questions. A student knows the correct answer to 14 questions and guesses at random for the other 6. Let X be the number of questions the student gets right. (a) Describe the distribution of X by a formula. (b) Give a numerical expression for P ( X 19). (c) Evaluate E ( X ) as a decimal. (d) Evaluate Var( X ) as a decimal. 3. Suppose X and Y are independent variables, such that X has uniform distribution on [0 , 3], and Y has exponential distribution with rate λ = 1. (a) Find P ( X < Y ). (b) Find the probability density function for Z := min( X,Y ), the mini- mum of X and Y .
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pitman_fall_2005_final - Statistics 134 Fall 2005 Final...

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