Let X be a zero- mean, unit- variance, Gaussian random variable and let Y be a chi-

Question:

Let X be a zero- mean, unit- variance, Gaussian random variable and let Y be a chi- square random variable with n€“1 degrees of freedom (see Appendix D, section D. 1.4). If X and Y are independent, find the PDF of

One way to accomplish this is to define an auxiliary random variable, U = Y, and then find the joint PDF of T and U using the 2 Ã— 2 transformation techniques outlined in Section 5.9. Once the joint PDF is found, the marginal PDF of T can be found by integrating out the unwanted variable U. This is the form of the statistic

Let X be a zero- mean, unit- variance, Gaussian random

Of Equation (7.41) where the sample mean is Gaussian and the sample variance is chi- square (by virtue of the results of Exercise 7.39) assuming that the underlying are Gaussian.