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Unformatted text preview: APPM 4/5520 Solutions to Problem Set Seven 1. The first population moment is μ 1 = E [ X ] = integraldisplay ∞ η x · e ( x η ) dx = η + 1 . (Note that since the pdf is that of an exponential with rate 1 that has been shifted by η to the right, the mean is 1 (the mean of the exponential with rate 1) that has been shifted by η to the right!) The first sample moment is M 1 = 1 n n summationdisplay i =1 X i = X. Equating them gives us η + 1 = X. Solving for θ gives us the MME: ˆ η = X − 1 . 2. This is a Beta distribution with a = θ and b = θ . The first distribution moment is μ 1 = E [ X ] = a a + b = θ θ + θ = 1 2 . This does not even involve θ so setting it equal to the sample moment M 1 = X will not get us anywhere where we can solve for θ . The second distribution moment is E [ X 2 ] = V ar [ X ] + ( E [ X ]) 2 = ab ( a + b ) 2 ( a + b +1) + parenleftBig 1 2 parenrightBig 2 = 1 4(2 θ +1) + 1 4 Setting the equal to M 2 = 1 n ∑ X 2 i and solving for θ gives ˆ θ MME = 1 4 n ∑ X 2 i 1 − 1 2 3. (a) You can get full credit here by just assuming a Poisson distribution. However, this problem is slightly different and I will give the solution for the stated problem... The only way we will observe a zero in the sample is if θ = 0. In the case where θ = 0, we are sampling values from the distribution given by f ( x ; 0) = P ( X = x  θ = 0) Since 1 = f (0; 0) = P ( X = 0  θ = 0) , when θ = 0 we will always observe the sampled X ’s to be zero. So, if we observe ( x 1 ,x 2 ,... ,x n ) = (0 , ,... , 0), we know that θ must have been zero. When θ > 0, we will be sampling X ’s from the distribution with pdf f ( x ; θ ) = θ x e θ x ! I { 1 , 2 ,... } ( x ) The likelihood is L ( θ ) ∝ n productdisplay i =1 f ( x i ; θ ) = n productdisplay i =1 θ x i e θ x i ! I { 1 , 2 ,... } ( x i ) = θ ∑ x i e nθ bracketleftBigg n productdisplay i =1 1 x i ! I { 1 , 2 ,... } ( x i ) bracketrightBigg Throwing out the constant of proportionality (wrt θ ), we may take the likelihood to be L ( θ ) = θ ∑ x i e nθ ....
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This note was uploaded on 02/07/2012 for the course APPM 4520 taught by Professor Manuel during the Fall '11 term at University of Colorado Denver.
 Fall '11
 Manuel

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