Question

and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ' using it, determine its corresponding distribution.

Answer 1

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Mx(t) (1-2t)-2 Proof Using the definition of moment generating function, we obtain Mx(t) Efexp(tX)] exp(bxfy(x)d exp(tx)x" exp /2- exp n2-1 2 dv (changing variabley= 1 exp ) 2t n/2 ynizexp=y}dy 21 ni2 2exp(-y)dy - 2t I(n/2) (by the definition of Gamrna function) (by the definition of c) 221(n/2) 1- 2 2n 2 1 2n2 (1-2r)2 (1 -2r)2 The integral above is well-defined and finite only when

Moment generating function The moment generating function of a normal random variable x is defined for any te R Mx(t) exp t+ Proof The mgf is derived as follows: (definition of mgf) Mx(t) E[exp(tX)] (where Z has a standard normal distr. - see above) - E[exp(t(4 + oZ))] exp(ut)E[exp(o1Z) (by linearity of the expected value = (Mz is the mgf of Z) - exp(ut)Mz(ot) (Han) (formula for the mgf of Z-see above) = exp(ut) exp = exp ut+ It is defined for any t e R because the moment generating function of z is defined for any teR.

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