A84 In class we showed that it X1 X2 Xn is a random sample

A.8.4. In class we showed that it X1, X2,.. ., Xn is a random sample from a normal distribution having mean sigma 1^2, then X-bar ~ Normal(mu1 /sigma1^2/n1 ). In this problem suppose Y1, Y2,...Yn2 are independent normally distributed random variables with mean mu2 and variance sigma2^2. Further suppose that the random sample of Y\'s is independent of the random sample of Xs, so that in particular, X-bar and Y-bar are independent random variables. Find the mgf of X-bar n1 - Yn2 to show that it has a normal distribution, and identify its mean and variance.

Solution

All xis are normal with mu1 and sigma1

All yis are normal with mu2 and sigma2

Consider xi-yj

As x and y are independent

Mean of xn1-yn2 = mu1 - mu2

Var(xn1-yn2) = var(xn1) + var(yn2) = sigma1^2+sigma2^2

Hence we get xn1-yn2 follow a normal distribution with mean = mu1 - mu2

and std deviation = rt of  where numerator = sigma1^2+sigma2^2

MGF of Xn1-yn2 = E(e^zt) = integral of

ext(e-(x-mu)^2/sigma^2dx/sigmart of 2pi

where x = xn1-yn2

mu = mu1-mu2

and sigma = rt of(sigma1^2+sigma2^2)