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. L

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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 L 1 (315.26 KiB) Viewed 33 times

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, zyn Un 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 h (2, 3) = PU, |Z = %), dx X, Z ER 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 e N, z"e-az- dz = r() Jo n+1 2a"