sol3 - HOMEWORK#3 SOLUTIONS IE121 7-19 f x = e − λx x...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: HOMEWORK #3 SOLUTIONS IE121 7-19. f ( x) = e − λ λx x! n L (λ ) = ∏ i =1 n n e λ e = xi ! xi −λ − nλ n λ n ∑ xi i =1 ∏x ! i i =1 ln L(λ ) = − nλ ln e + ∑ x i ln λ − ∑ ln x i ! i =1 i =1 d ln L(λ ) 1 = −n + dλ λ n ∑x i =1 i n i ≡0 −n+ ∑x i =1 λ =0 ∑x i =1 n i = nλ ˆ λ= 7-26. ∑x i =1 n i n a) Because a is less than or equal to "a" for every sample, E( a ) cannot equal "a". n n b) Yes, E( a ) is less than "a" by a factor of . As n → ∞, → 1, and E( a ) → a. n+1 n+1 (n +1) , because Ea (n +1) = n + 1E(a) = a . c) a n n a d) FY(y) = P(Y≤ y)=P(X1≤ y, X2≤ y,..., Xn≤ y) = P(X1≤ y)P(X2≤ y)...P(Xn≤ y) = (y/a)n for 0 ≤ y ≤ a. f ( y) = ∂FY ( y) nyn−1 = for 0≤y≤a ∂y an =0 otherwise The maximum likelihood estimator for a is Y. To show that the mle for a is biased, need to show that E(Y) ≠ a: E( Y ) = ∫ y 0 a nyn−1 an = nyn an (n + 1) a = 0 n a. n+1 7-35. Thus, E(Y) ≠ a, and the mle of a is biased. σ 3.5 = = 1.429 , µ X = 75.5 psi σ X = n 6 75.75 − 75.5 1.429 X −µ P ( X ≥ 75.75) = P ≥ σ / n = P ( Z ≥ 0.175) = 1 − P ( Z ≤ 1.75) = 1 − 0.56945 = 0.43055 7-51. X ~ N (50,144) 53− 50 12 / 36 P (47 ≤ X ≤ 53) = P 47 −50 ≤ Z ≤ 12 / 36 = P (−1.5 ≤ Z ≤ 1.5) = P ( Z ≤ 1.5) − P ( Z ≤ −1.5) = 0.9332 − 0.0668 = 0.8664 7-57. V ( X ) = V [ aX 1 + (1 − a ) X 2 ] = a 2V ( X 1 ) + (1 − a ) 2V ( X 2 ) = a 2 ( σ ) + (1 − 2a + a 2 )( σ ) n n 1 2 2 2 = a 2σ 2 σ 2 2aσ 2 a 2σ 2 + − + n2 n2 n2 n1 2 = (n2 a 2 + n1 − 2n1a + n1a 2 )( σ ) n1n2 2 ∂V ( X ) = ( σ )(2n2 a − 2n1 + 2n1a ) ≡ 0 n1n2 ∂a 0 = 2n2 a − 2n1 + 2n1a 2a ( n2 + n1 ) = 2n1 a ( n2 + n1 ) = n1 n1 a= n2 + n1 n −1 i =1 7-70. E (V ) = k ∑ [ E ( X i2+1 ) + E ( X i2 ) − 2 E ( X i X i +1 )] = k ∑ (σ 2 + µ 2 + σ 2 + µ 2 − 2 µ 2 ) i =1 n −1 Therefore, k = = k ( n − 1)2σ 2 1 2 ( n −1) ...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

sol3 - HOMEWORK#3 SOLUTIONS IE121 7-19 f x = e − λx x...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online