I have a random variable $X$ following the gaussian distribution $N(\mu, \sigma^2)$. If I define $Y := X^2$, then I know that $Y$ is has a non central chi square distribution with 1 degree of freedom : $$ Y \sim \chi^2_1(\lambda), \ \ \lambda = \frac{\mu^2}{\sigma^2}, $$ right?
Now if I define a third variable $$ Z := Y + k = X^2 + k, $$ with $k$ a positive constant, what is the distribution of $Z$? Is it also noncentral chi square? If so, with which parameter $\lambda$?