Transformation of any normal distribution into a standardized t-distribution

Question

What will be the transformed Mean and transformed standard deviation if any normal distribution is transformed into a standardized t-distribution?

Does t force the mean and standard deviation to be 0 and 1 respectively?

Answer 1

$$\frac{N(0,1)}{\sqrt\frac{\chi_v^2}{v}}\sim t_v$$transformed mean will be 0 and transformed standard deviation will be $\sqrt\frac{v}{v-2}$ which is same as standard t-distribution's.

Proof