Be $X\sim N(\mu,1)$ and $Y\sim Inverse-Gamma(\alpha,\beta)$.
I need to find the distribution of $T=X\sqrt{Y}$.
According to my calculations, T is not a non-central Student's t-distribution , it is a non-standardized Student's t instead with $2\alpha$ degrees of freedom, location parameter $\mu$ and a scale parameter $\sqrt{\frac{\beta}{\alpha}}$.
Is it correct?
Thank you.