The usual t distribution, as derived in Section 5.3.2, is also known as a central t distribution. It can be thought of as the pdf of a random variable of the form T = n(0,1)/ˆšX2v/v, where the normal and the chi squared random variables are independent. A generalization of the t distribution, the noncentral t. is of the form Tʹ = n(μ, 1) / ˆšX2v/v, where the normal and the chi squared random variables are independent and we can have μ ‰  0. (We have already seen a noncentral pdf, the noncentral chi squared, in (4.4.3).) Formally, if X ~ n(μ, 1) and Y ~ X2v, independent of X, then Tʹ = X/ˆšY/v has a noncentral t distribution with v degrees of freedom and noncentrality parameter δ =ˆšÎ¼2.
(a) Calculate the mean and variance of Tʹ.
(b) The pdf of Tʹ is given by

Show that this pdf reduces to that of a central t if δ = 0.
(c) Show that the pdf of Tʹ has an MLR in its noncentrality parameter.

rr 2/v)*/2(5t)* _r{[v + k+ 1]/2) (1+ (22/v))(u+k+1)/2 * e-/a fr (t6) = k! k-0