hw7sol - APPM 4/5520 Solutions to Problem Set Seven 1. The...

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Unformatted text preview: APPM 4/5520 Solutions to Problem Set Seven 1. The first population moment is 1 = E [ X ] = integraldisplay x e- ( x- ) dx = + 1 . (Note that since the pdf is that of an exponential with rate 1 that has been shifted by to the right, the mean is 1 (the mean of the exponential with rate 1) that has been shifted by to the right!) The first sample moment is M 1 = 1 n n summationdisplay i =1 X i = X. Equating them gives us + 1 = X. Solving for gives us the MME: = X 1 . 2. This is a Beta distribution with a = and b = . The first distribution moment is 1 = E [ X ] = a a + b = + = 1 2 . This does not even involve so setting it equal to the sample moment M 1 = X will not get us anywhere where we can solve for . 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 +1) + 1 4 Setting the equal to M 2 = 1 n X 2 i and solving for gives 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 = 0. In the case where = 0, we are sampling values from the distribution given by f ( x ; 0) = P ( X = x | = 0) Since 1 = f (0; 0) = P ( X = 0 | = 0) , when = 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 must have been zero. When > 0, we will be sampling X s from the distribution with pdf f ( x ; ) = x e- x ! I { 1 , 2 ,... } ( x ) The likelihood is L ( ) n productdisplay i =1 f ( x i ; ) = n productdisplay i =1 x i e- x i ! I { 1 , 2 ,... } ( x i ) = x i e- n bracketleftBigg n productdisplay i =1 1 x i ! I { 1 , 2 ,... } ( x i ) bracketrightBigg Throwing out the constant of proportionality (wrt ), we may take the likelihood to be L ( ) = x i e- n ....
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hw7sol - APPM 4/5520 Solutions to Problem Set Seven 1. The...

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