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) X2v v, where the normal and the chi squared random variables are independent A generalization of the t distribution, the noncent...

a We will use the equality in Exercise 317 which re ...

The usual t distribution, as derived in Section 5.3.2, is also known as a central t distribution.

Question:

The usual t as derived in Section 5.3.2, is also known as a central t 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 the noncentral t. is of the form TÊ¹ = n(Î¼, 1) / ˆšX2v/v, where the normal and the chi squared random variables are independent and we can have Î¼ ‰ 0. (We have already seen a noncentral pdf, the noncentral chi squared, in (4.4.3).) Formally, if X ~ n(Î¼, 1) and Y ~ X2v, independent of X, then TÊ¹ = X/ˆšY/v has a noncentral t with v degrees of freedom and noncentrality parameter Î´ =ˆšÎ¼2.
(a) Calculate the mean and variance of TÊ¹.
(b) The pdf of TÊ¹ is given by

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

Show that this pdf reduces to that of a central t if Î´ = 0.
(c) Show that the pdf of TÊ¹ has an MLR in its noncentrality parameter.

Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...