1. The random variable X has mean u and variance sigma^2. The random variable Y is defined as Y=g(x).
(a) Use a first order Taylor Series expansion about u to prove the following (see attachment).
(b) If X follows the exponential distribution with mean u, so sigma^2 = u^2, and Y = g(x) = sqrt(x), use the results in the question above to prove the following (see attachment).
(c) Use the method of transformation to find the probability density function for Y = sqrt(X) and show that it can be written in the form of a Weibull distribution with probability density function (see attachment).
(d) Give that the Weibull distribution stated above has a mean and variance that is defined in the attachment, compare the approximations for the mean and variance of Y with their true values.
The solution gives detailed steps on finding the probability density function of a new variable using method of transformation and determining its mean and variance.