A question in addition to: Cauchy and $\chi^{2}$ dist
I have to find the distribution of $Z = Y\sqrt X$
I know Y is given by a fraction of 2 standard normal random variables ($Y = \frac{V_{1}}{V_{2}}$) and $\sqrt X = V_{2}$ is a standard normal random variable and therefore I conclude that $Z = \frac{V_{1}}{V_{2}}\cdot V_{2} = V_{1}$ is standard normal distributed.
Is this correct??