PLEASE SHOW ALL WORK AND STEPS Thanks Let X1x5 NmU1 sigma12

Let X_1,...,x_5 N(mU_1, sigma1^2). Let Y1,... ,Y5 N (mU_2, sigma ^2 2). Also let all the Xs and Ys be independent, Recall that we may write a chi-square random variable as follows: where Zx.....Zv N(0,1). Use this and other properties about normal rvs to prove that both sigma 5 i = 1 (xi - mu 1 / sigma 1)^2 and sigma 5 i = 1 (Yi - mU_2 / sigma2)^2 follow a x^2 5 distribution. Recall that the moment generating function of chi-square rv with u degrees of freedom is (1 - 2t)^-v / 2 . Use this to prove that 5 sigma i = 1 (Xi - mu 1 / sigma1)^2 + 5 sigma i = 1 (Yi - mU_2 / sigma)^2 follows a chi-square distribution with ten degrees of freedom chi^2 10. Recall that an Fa.h may be written as F a,b = chi^2 a / a / chi^2 b / b use this to show that sigma^5 i = 1 (xi - mu 1 / sigma1)^2 / sigma 5 i = 1 (Yi - mU_2 / sigma2)^2 F5.5

Solution

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