hw5 - P ( M > m + n | M > m ) = (1-p...

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MATH 471 HW 5 Solutions Pengsheng Ji 3.1.11 We know Poisson distribution can be used to approximate the binomial distribu- tion. In this problem, p=0.01, n=25, λ = np = 0 . 25. Using Poisson approximation, we have P ( X = 0) + P ( X = 1) = e - 0 . 25 + e - 0 . 25 · 0 . 25 1 1! = 0 . 9735 . 3.2.7 If f ( x ) is a density function, then Z -∞ f ( x ) dx = Z 3 - 3 f ( x ) dx = 1 . Thus, 1 = Z 3 - 3 f ( x ) dx = c Z 3 - 3 (3 - | x | ) dx = 9 c, which gives c=1/9. 3.2.13 The density of the exponential distribution is f ( x ) = ( λe - λx x 0 0 otherwise Then, P ( X > 2 ) = 1 - F (2 ) = 1 - (1 - Z 2 0 λe - λx dx ) = 1 - (1 - e - 2 ) = e - 2 . 3.2.17 Note P ( M = j ) = (1 - p ) j p and then P ( M k ) = X j = k (1 - p ) j p = (1 - p ) k . Or we can get this result more directly since M k if and only if the first k trials result in failure. Then P ( M m + n | M m ) = (1 - p ) m + n (1 - p ) m = (1 - p ) n = P ( M n ) .
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Unformatted text preview: P ( M > m + n | M > m ) = (1-p ) m + n +1 (1-p ) m +1 = (1-p ) n 6 = (1-p ) n +1 = P ( M > n ) . 3.2.19 (a) ( t ) = lim h 1 h P ( t < T < t + h ) P ( T > t ) = lim h R t + h t f ( x ) dx h 1 1-F ( t ) = f ( t ) 1-F ( t ) . To get the last result, we were using the following theorem in Calculus: Theorem If f(t) is continuous at t, then lim h R t + h t f ( x ) dx h = f ( t ) . (b) Using the nal result in (a) we can get ( t ) = e-t /e-t = (c) ( t ) = 2 te-t 2 e-t 2 = 2 t...
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hw5 - P ( M > m + n | M > m ) = (1-p...

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