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Let X1 ,X2 be a random sample of size 2 from a distribution with positive variance and m.g.f. M( t). If Y = X1 + Xi and Z X1 ·- X2 are independent, prove that the distribution from which the sample is taken is a normal distribution.

Hint: Show that m( t., ti ) = E{exp [t1 (X1 + X2 ) + 12 (X1 - X2 )I} = M(11 + 12 )M( t 1 - t2 ).

Express each member of m(11 • 12 ) = m(11 • O)m(O. 12 ) in terms- of M; differ entiate twice with respect to 12 ; set 12 = 0; and solve the resulting differential equation in M.

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Statistics and Probability, Statistics

Category:- Statistics and Probability
Reference No.:- M91702024

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