Chi square distribution.

Question

Let $X$ be a random variable which follows $N(0,1)$. It is clear that $X^2$ follows $\chi^{2}(1)$. Let $k\in\mathbb{R}^{+}$ be a positive real number. I want to know the distribution of the random variable $\sqrt{X^{2} + k}$.

Thanks in adnvance.

Answer 1

You can start defining analytically the density of $Y=X^2$ that is

$$f_Y(y)=\frac{1}{\sqrt{2\pi y}}e^{-y/2}$$

$y>0$

and then proceed by transforming your rv in the desired $Z=\sqrt{Y+k}$

It results to me

$$f_Z(z)=\frac{2z}{\sqrt{2\pi}\sqrt{z^2-k}}e^{-(z^2-k)/2}\cdot\mathbb{1}_{(\sqrt{k};+\infty)}(z)$$

this is the density of your rv.