s330probf08

s330probf08 - STATISTICS 330 PROBLEMS 1 Consider the...

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STATISTICS 330 PROBLEMS 1. Consider the following functions: (a) f ( x )= kx (0 . 3) x ,x =1 , 2 ,... (b) f ( x 2 , 2 (c) f ( x k (1 + x 2 ) 1 , −∞ <x< (d) f ( x ke | x | , −∞ (e) f ( x k (1 x ) 5 , 0 1 (f) f ( x 2 e λ x , x> 0 , λ > 0 (g) f ( x ( θ +1) , x α > 0 , θ > 0 (h) f ( x x/ θ ¡ 1+ e x/ θ ¢ 2 , −∞ , θ > 0 (i) f ( x 3 e 1 / ( θ x ) > 0 , θ > 0 In each case: ( α ) Determine k so that f ( x ) is a p.f./p.d.f. and sketch f ( x ) . ( β ) Let X be a random variable with p.f./p.d.f. f ( x ) . Find the c.d.f of X . ( γ ) Find E ( X ) and Var ( X ) . ( δ ) Find P (0 . 5 <X 2) and P ( X> 0 . 5 | X 2) . To obtain numerical answers for ( f ) ( i ) use λ for ( f ) ,use α = θ for ( g ) , use θ =2 for ( h ) and use θ for ( i ) . 2. Suppose X v GEO ( p ) . (a) Show that P ( X k + j | X k P ( X j ) where k and j are nonnegative integers. Explain why this is called the memoryless property. (b) The only other distribution with this property is the exponential distribution. Show that Y v EXP( θ ) satis f es the memoryless property. 3. (a) If X v GAM( α , β ) then f nd the p.d.f. of Y = e X . (b) If X v GAM( α , β ) then show Y = X 1 IG ( α , β ) . (c) If X v N ¡ μ , σ 2 ¢ then f nd the p.d.f. of Y = e X . (d) If X v N ¡ μ , σ 2 ¢ then f nd the p.d.f. of Y = X 1 . (e) If X v UNIF( π 2 , π 2 ) then show that Y =tan X v CAU(1 , 0) . (f) If X v PAR ( α , β ) then show that Y = β log( X/ α ) v EXP(1) . (g) If X v DE (1 , 0) then f nd the p.d.f. of Y = X 2 . (h) If X v t( k ) then show that Y = X 2 v F(1 ,k ) . 4. Suppose that f 1 ( x ) ,f 2 ( x ) ,...,f k ( x ) are p.d.f.’s with supports A 1 ,A 2 ,...,A k , means μ 1 , μ 2 ,..., μ k , and f nite variances σ 2 1 , σ 2 2 σ 2 k respectively. Suppose also that 0 p 1 ,p 2 ,...,p k 1 and P k i =1 p i . Show that g ( x P k i =1 p i f i ( x ) is a p.d.f. . Let X be a random variable with p.d.f. g ( x ) . Find the support of X , the mean of X and the variance of X . 5. If E ( | X | k ) exists for some k Z + then show that E ( | X | j ) exists for j ,...,k 1 . 6. Suppose T v t ( n ) . (a) Show that E ( T )=0 if n> 1 . (b) Show that ( T n/ ( n 2) if 2 . 1
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7. For each of the following p.f./p.d.f.’s derive the moment generating function M ( t ) . State the values for which M ( t ) exists and use the m.g.f. to f nd the mean and variance. (a) f ( x )= ¡ n x ¢ p x (1 p ) n x , x =0 , 1 ,...,n ;0 <p< 1 (b) f ( x μ x e μ /x ! , x , 1 ,... ; μ > 0 (c) f ( x 1 β e ( x θ ) / β ,x > θ ; −∞ < θ < , β > 0 (d) f ( x 1 2 e | x θ | , −∞ <x< ; −∞ < θ < (e) f ( x )=2 x , 0 1 (f) f ( x x 0 x 1 2 x 1 <x 2 0 otherwise 8. Suppose X is a random variable with m.g.f. M ( t E ( e tX ) which exists for t ( h,h ) for some h> 0 . Then K ( t )=log M ( t ) is called the cumulant generating function (c.g.f.) of X. (a) Show that E ( X K 0 (0) and Var ( X K 00 (0) . (b) If X v NB( k,p ) then use ( a ) to f nd E ( X ) and ( X ) . 9. For each of the following f nd all the moments of X if X is a random variable with m.g.f. M ( t ): (a) M ( t )=(1 t ) 3 , | t | < 1 (b) M ( t )=(1+ t ) / (1 t ) , | t | < 1 (c) M ( t e t / (1 t 2 ) , | t | < 1 10. Suppose Z v N(0 , 1) and Y = | Z | . (a) Show that M Y ( t Φ ( t ) e t 2 / 2 , −∞ <t< , where Φ ( t ) is the c.d.f. of a , 1) random variable.
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This note was uploaded on 04/27/2010 for the course STAT 330 taught by Professor Paulasmith during the Fall '08 term at Waterloo.

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s330probf08 - STATISTICS 330 PROBLEMS 1 Consider the...

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