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 × 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
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.
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