STAT 400 Lecture AL1 1 Let 0 and let X 1 X 2 X n be a random sample from the distribution with the probability density function f X x f X x a Find E X k k E Xk xk 0 u 0 b Spring 2015 Dalpiaz Examples for 6 4 Part 3 1 2 x 2 2k e u x 2 e Hint Consider u e du x dx x 0 x u x 1 x u 2k 2k 0 e u du Find the method of moments estimator of x 1 0 01 x 2 0 04 x 3 0 09 x 4 0 36 1 2k du 2 x dx 2 k 1 Suppose n 4 and Find the method of moments estimate of E X E X1 X n X n i 2 1 i 1 2 1 2 3 2 2 n X i i 1 0 50 x 0 125 2 2 2 X 2 n n X i i 1 2 0 125 4 OR E X x f x dx x 2 0 x 0 y e x dx dy x dx 2 x E X y 2 e y dy 0 Integration by parts Choice of u b u dv u v a u y 2 E X y e 2 u y b a L A T E b v du a dv e y dy y 0 2 y e y ogarithmic lgebraic rigonometric xponential v e y du 2 y dy dy 2 y e y dy 0 0 dv e y dy du dy v e y 2 1 y 1 y E X 2 y e e dy e y dy 0 0 0 2 1 2 e y 2 0 OR 1 0 0 x f x dx x E X x 2 y E X y 2 e x dx dy x dx 2 x e y dy E Y 2 0 where Y has Exponential distribution with mean 2 E X E Y Var Y E Y c 2 1 1 2 1 Find the maximum likelihood estimator of x 1 0 01 x 2 0 04 x 3 0 09 x 4 0 36 2 2 2 Suppose n 4 and Find the maximum likelihood estimate of n L i 1 2 x i e n ln L n ln ln 2 i 1 ln L n xi n xi i 1 xi n i 1 xi 0 n n i 1 n i 1 x i 1 2 Xi 4 10 3 3333 3 1 2 2 Let 0 and let X 1 X 2 X n be a random sample of size exponential distribution That is f x a 2 e x from a double x Find E X k for positive integer k E Xk k odd xk 2 e x 0 dx xk 2 e x dx xk 0 2 e x dx 2 0 xk 2 k 1 k 0 e x dx k x x e dx 0 k 1 k 1 1 x k 1 k x dx e k k 1 k Obtain the maximum likelihood estimator of L 0 k even b n n n exp xi 2n i 1 d n n ln L xi 0 d i 1 n ln L n ln n ln 2 xi i 1 n n i 1 Xi 3 Let 0 and let X 1 X 2 X n be a random sample from the distribution with the probability density function f x 3 x 5 e x 2 a Find E X k k 6 E Xk x 0 Hint Consider u x 2 k 3 5 x dx x x e 2 u x2 du 2 x 0 k 2 1 2 u 2 u e du k 2 u 2 k 2 e u du 2 2 0 0 k 1 k 2 3 2 2 b Obtain the method of moments estimator of x 1 4 x 2 2 x 3 4 x 4 3 Suppose n 4 and Find the method of moments estimate of E X 15 16 1 1 2 5 3 1 1 1 2 5 3 1 1 2 2 2 2 2 2 2 2 2 X x 1 4 1 1 2 1 1 1 1 2 5 1 1 2 5 3 1 3 1 2 2 2 2 2 2 2 2 2 2 256 X 2 225 2 15 16 256 X x 2 2 225 x 3 4 x 4 3 225 0 2614 2704 x 13 3 25 4 OR 1 1 3 3 1 2 2 2 4 3 2 2 2 2 E X2 3 n 1 X2 X i2 2 n i 1 3 X2 3n n X i2 i 1 n x 1 4 x 2 2 2 3n n x 3 4 x i2 x 4 3 45 i 1 12 4 0 2667 45 15 x i2 i 1 c Obtain the maximum likelihood estimator of x 1 4 x 2 2 x 3 4 Suppose n 4 and x 4 3 Find the maximum likelihood estimate ln L 3 n ln ln x i5 x i2 of L n 3 x i5 e i 1 ln L x i2 n x i2 0 i 1 3n n i 1 3n n X i2 i 1 n x 1 4 x 2 2 3n n x i2 i 1 x 3 4 x 4 3 12 4 0 2667 45 15 x i2 i 1 45 n i 1 Useful facts Def x u x 1 e u du x 0 0 1 1 x x 1 x 1 n n 1 1 2 if n is an integer
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