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Unformatted text preview: Probability: Final Exam Solutions Instructor: W. D. Gillam Part I. This material is in the text or your notes. (1) (3 Points) Give the probability P ( X = k ) when X is a discrete random variable with a.... (a) binomial distribution based on n trials with success probability p . (b) Poisson distribution with expected value . (c) negative binomial distribution counting the number of success probability p trials up to the n th success. (2) (3 Points) Give the definitions of the beta and gamma functions, explain how they are related, and give the functional equation for the gamma function. (3) (4 Points) Give the density functions for the gamma distribution and the beta dis tribution (with parameters , ) and give the expected value of each distribution. Give the cumulative distribution function for the exponential distribution. Solution. Many of you seem to have forgotten the CDF: F ( x ) = Z x e y/ dy = h e y/ i x = 1 e x/ . (4) (3 Points) Give Stirlings approximation of n ! and name two places where we used it. Solution. I was hoping you would mention the de MoivreLaplace Theorem (the Central Limit Theorem for Bernoulli trials) and Polyas theorem on random walks. (5) (3 Points) Give the definition of the n th harmonic number H n and say something intelligent about the harmonic numbers. Solution. H n = 1+1 / 2+ +1 /n . I was hoping you would mention the recursion from your homework, or the generating function X n H n x n = ln(1 x ) 1 x , or perhaps the coupon collecting problem or the Markov chain problem whose solutions involved the harmonic numbers. 2 (6) (3 Points) Give Chebyshevs two bounds and state the Weak Law of Large Num bers. (7) (2 Points) State the Central Limit Theorem. Part II. (8) (5 Points) Suppose X,Y are independent continuous random variables with den sity functions f ( x ) = x/ 2 , x 2 , otherwise g ( x ) = 3 x 2 , x 1 , otherwise (respectively). Calculate E ( X ) and E ( Y ). Calculate the density function for X + Y and use it to calculate E ( X + Y ). Solution. E ( X ) = R 2 x ( x/ 2) = [ x 3 / 6] 2 = 4 / 3 and E ( Y ) = R 1 3 x 3 = [3 x 4 / 4] 1 = 3 / 4 . The density h ( x ) for X + Y is given by convolution: h ( x ) = Z R f ( y ) g ( x y ) dy. The integrand is zero unless 0 y 2 and 0 x y 1 ( i.e. unless 0 y 2 and x 1 y x .) When 0 x 1, these two constraints are equivalent to y x hence h ( x ) = Z x ( y/ 2)3( x y ) 2 dy = Z x 3 2 y 3 3 xy 2 + 3 2 x 2 y dy = 3 8 y 4 xy 3 + 3 4 x 2 y 2 x = (1 / 8) x 4 . When 1 x 2, the two constraints are equivalent to x 1 y x and we have h ( x ) = Z x x 1 ( y/ 2)3( x y ) 2 dy = 3 8 y 4 xy 3 + 3 4 x 2 y 2 x x 1 = 1 8 x 4 3 8 y 4 xy 3 + 3 4 x 2 y 2 fl fl fl fl y = x 1 = 1 8 x 4 3 8 ( x 1) 4 + x ( x 1) 3 3 4 x 2 ( x 1) 2 = x/ 2 3 / 8 . 3 When 2 x 3, the constraints are are equivalent to x 1 y 2 and we have h...
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This note was uploaded on 04/24/2011 for the course ECE 6303 taught by Professor Voltz during the Spring '11 term at NYU Poly.
 Spring '11
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