= Solution
Conditional on $\mu,\psi>0$, let $\lambda\sim\chi^2_4$ and independently $Z\sim N(0,1)$. The <normal distribution> with the stated <precision parameter> can be generated as
$$
\beta_j=\mu+\frac{2\psi Z}{\sqrt\lambda}.
$$
Therefore
$$
\boxed{\frac{\beta_j-\mu}{\psi}=\frac Z{\sqrt{\lambda/4}}\sim t_4,}
$$
by the defining <normal distribution> and <chi-squared distribution> representation of <Student's t-distribution>. Its density is
$$
f(\beta_j\mid\mu,\psi)=\frac3{8\psi}\left(1+\frac{(\beta_j-\mu)^2}{4\psi^2}\right)^{-5/2}.
$$
Thus \b[$\mu$ is the location and $\psi$ is the scale], not the <standard deviation>: $\operatorname{Var}(\beta_j\mid\mu,\psi)=2\psi^2$. Each study gets its own independent chi-squared draw in this <Student t random-effect model>.
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