75. If 9..., are independent unit normal random variables, then X = ?= is s ai d...

75. If ÆÀ9...,Æç are independent unit normal random variables, then X = Ó?=\ÆÀ is s ai d to be a chi-square random variable with ç degrees of freedom. Compute its moment generating function.

(i) P[a record occurs at time ç} = l/n (ii) ^[number of records by time ç] = Óº=é 1/'
(iii) Var(number of records by time ri) = Ó?=é(' ~~ I ) / ' 2 (iv) Let TV = min[/?: ç > 1 and a record occurs at time n}. Show Å [Í] = oo.
Hint: For (ii) and (iii) represent the number of records as the sum of indicator (that is, Bernoulli) random variables.