Suppose $Z\sim N(\mu,1)$ and $V$ is independent of $Z$ with distribution $\chi^2_m$. Then $T=\frac{Z}{(V/m)^{1/2}}$ is said to have a noncentral $t$ distribution with noncentrality $\mu$ and $m$ degrees of freedom. We deduced that $$P(T\leq t)=2m\int_0^\infty \Phi(tw-\mu)f_m(mw^2)wdw$$ where $fm(w)$ is the $\chi^2_m$ density, and $\Phi$ is the normal distribution function.

It is also evident that $T^2$ has a noncentral $F_{1,m}$ distribution with noncentrality parameter of $\mu^2$. Now, what I want to derive is the density of $T$, which is in a form of: $$p(t)=2\sum_{i=0}^\infty P(R=i).f_{2i+1}(t^2)[\phi(t-\mu)I(t>0)+\phi(t+\mu)I(t<0)]$$ where $R\sim {\cal{P}}(\frac{1}{2}\theta^2)$. One thing that I thought might be beneficial to do is to condition it on $|T|$ first, but I wasn't able to get $p(t)$.

Just to clarify a bit on the structure of the pdf, I have found these: enter image description here enter image description here


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