Chi-squared distribution multiplied by constant

Question

I have a question regarding notation.

When $\chi^2(p)$ is the chi-squared distribution with degrees of freedom $p$, then what does it mean that $$ X \sim k \chi^2(p), $$ where $k$ is some constant? I mean what distribution is $k \chi^2(p)$?

Answer 1

It means $X=kY$ with $Y\sim\chi^2(p)$.

Answer 2

$\chi^2(p)$ is the distribution of the sum of the squares of p independent standard normals. I doubt that $k\chi^2(p)$ has its own name. If $y = kx \wedge x \sim \chi^2(p)$. You can use $P(y \leq z) = P(x \leq \frac{z}{k})$ to obtain the distribution.