Question

Proof of it please..

Theorem 2. The PDF of T defined in (7) is given by fa() Proof. The proof is left as an exercise.

Answer 1

Since, the given PDF is of Student's t- distribution where 'n' is degree of freedom and $\Gamma$ is the Gamma Function.

Let Z be a random varaiable that follows N(0,1) i.e. with mean '0' and Variance '1' and chi-square distributed independently of Z.

Now, Applying Jacobian transformation and then using differential equation:

So, obtaining the marginal distribution of 't' would be as:

Next, we would obtain the integral function of General Gamma Distribution,which are as follows:

therefore, doing some calculations we got our result.

Hence, we have proved the above PDF(Probability Density Function).