Un Problem 2* The aim of this exercise is to deduce the p.d.f of the Student's t distribution t(n) discussed in lectures

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Un Problem 2 The Aim Of This Exercise Is To Deduce The P D F Of The Student S T Distribution T N Discussed In Lectures 1 (316.21 KiB) Viewed 36 times

Un Problem 2* The aim of this exercise is to deduce the p.d.f of the Student's t distribution t(n) discussed in lectures. Let Wn ~ x?(n) be a chi-square random variable with n degrees of freedom and let Z ~ N(0,1) be a standard normal random variable with Z and Wn independent. Then we define the following random variable, Zn W. and say that Un~t(n). (a) Using the p.d.f. of Wn given in lectures, find a formula for the conditional density hn(x, 2) = PUR e <= ) -P(Un < x | Z = z), X, Z ER dx Hint: be careful with the sign of x and z. Start by assuming x > 0 and compute hn(x,z) separately for cases z < 0 and z > 0. Then deal with the case x < 0 similarly. (b) Show that for any a > 0 and n € N, z"e-az- dz r("72) 6.**** 1+1 2a"