= 20- fr(y)= yv2πσ 2. Let X be a normal random variable with finite mean u and variance o?. It is known that the probabi

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= 20- fr(y)= yv2πσ 2. Let X be a normal random variable with finite mean u and variance o?. It is known that the probability density function fx(x) of X and its moment generating function Mx():= E(COX) are given by fx(x) = vzto exp(-) and Mx (*) = exp(48+02), respectively. Next, define Y = ek, which is often referred to as the log-normal random variable. Y (a) Show that the random variable Y has the probability density function given by (logy- - „V2roz exp ( - 10?). 120. 202 () Show that the mean and variance of Y are given subsequently by E(Y) = exp(+ to and Var(Y) = exp(24 +0°) (exp(0%) - 1). (c) We are interested in estimating the distribution parameters u and gº from independent observation Y,,...,Y, of the random variable Y. i. Write down the log-likelihood function of the observation Y.....,Y. ii. Show that the maximum likelihood estimations of u and o are given by û=- Ülogy, and 72 = (logy; – )? (1) ha (d) Following the two estimators of u and o given in (1), i. show that û is unbiased estimator of , whereas @ is a biased estimator of o?. ii. Show that S = = (logy - A)is unbiased estimator of 02. iii. Use the following observation of Y to give (unbiased) estimate of u and o? 9.172 6.083 8.414 6.550 5.969 6.594 7.534 7.714 7.372 6.879 7.615 7.979 Note that n = 12, Îlog Y = 23.8 and (log y)2 = 47.386 Hint: Define for a given zi, i = 1,...,1,I= { 2. Use the fact that 1 i=1 i=1 i=1 1 (21-7) Σε -2. n (e) Show that the estimates u, 22 given in (1) and S2 defined in 2(d)ii are consistent estima- tors in the weak sense of the respective parameter