Dr. Hackney STA Solutions pg 25

Dr. Hackney STA Solutions pg 25 - 2-10 n Solutions Manual...

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Unformatted text preview: 2-10 n Solutions Manual for Statistical Inference = y=1 n a n-1 (y - 1) + (a + 1) y - 1 n a+b-1 a (n-1)+(a+1)+b-1 (y-1)+(a+1) a+1+b-1 a+1 (n-1)+(a+1)+b-1 (y-1)+(a+1) = na a+b-1 a+1 a a+1+b-1 a+1 n-1 a+1 (y - 1) + (a + 1) y - 1 y=1 = na a+b n-1 j=0 a+1 n-1 j + (a + 1) j a+1+b-1 a+1 (n-1)+(a+1)+b-1 (j+(a+1) = na , a+b since the last summation is 1, being the sum over all possible values of a beta-binomial(n - 1, a + 1, b). E[Y (Y - 1)] = n(n-1)a(a+1) is calculated similar to EY, but using the identity (a+b)(a+b+1) y(y - 1) n = n(n - 1) n-2 and adding 2 instead of 1 to the parameter a. The sum over all y y-2 possible values of a beta-binomial(n - 2, a + 2, b) will appear in the calculation. Therefore Var(Y ) = E[Y (Y - 1)] + EY - (EY )2 = 2.30 a. E(e tX tX nab(n + a + b) . (a + b)2 (a + b + 1) )= )= b. E(e c. c c tx 1 1 1 1 1 e c dx = ct etx 0 = ct etc - ct 1 = ct (etc 0 c 2x tx e dx = c22t2 (ctetc - etc + 1). 0 c2 - 1). (integration-by-parts) 1 -(x-)/ tx e e dx 2 1 e/ 1 e-x( -t) 1 2 ( - t) E(e ) tx = - 1 (x-)/ tx e e dx + 2 1 1 ex( +t) 1 ( +t) = = d. E etX = that e -/ 2 +- - 4et , 4- 2 t2 -2/ < t < 2/. r+x-1 x=0 x r tx r+x-1 x=0 e x pr (1 - p)x = pr t x (1 - p)e t x . Now use the fact r+x-1 x=0 x (1 - p)e 1 - (1 - p)e p 1-(1-p)et t r = 1 for (1 - p)et < 1, since this is just the sum of this pmf, to get E(etX ) = , t < - log(1 - p). 2.31 Since the mgf is defined as MX (t) = EetX , we necessarily have MX (0) = Ee0 = 1. But t/(1 - t) is 0 at t = 0, therefore it cannot be an mgf. 2.32 d S(t) dt = t=0 d (log(M x (t)) dt = t=0 d dt Mx (t) Mx (t) = t=0 EX = EX 1 since MX (0) = Ee0 = 1 d2 S(t) dt2 = t=0 d dt Mx (t) Mx (t) = t=0 2 Mx (t)M x (t) - [M x (t)] [M x (t)] 2 2 t=0 = 2.33 a. MX (t) = EX = tx e- x x=0 e x! 1 EX 2 -(EX) 1 = e- t = VarX. = e- ee = e(e t t (et )x x=1 x! -1) . d dt Mx (t) t=0 = e(e -1) et t=0 = . ...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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