A question on Noncentral Chi-squared distribution

[hntko]

Dear friends,

I have a short question on Noncentral Chi-squared distribution, and I may need your help.

If X is distributed according to the noncentral chi-squared distribution [*], so what is distribution of cX (given that c > 0)?
I tried to find the answer on Google but I have not seen it yet. I hope you may read it somewhere, or you may have a comment on this.

Thank you so much.

[*] https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

 

[Mityan]

Chi-squared in general is a particular case of gamma distribution. Look at scaling property of the last one on wiki.

 

[Shug]

Noncentral chi-squared... I don't envy you, those are messy!

You can answer this question via probability theory. If Y=g(X), then
\[f_Y = \frac{f_X(g^{-1})}{g'(g^{-1})},\]
which in the scalar case simplifies to
\[f_Y = \frac{f_X(y/c)}{c}.\]

 

[hntko]

Dear friends,

Thank you so much. With your comments, I'll try to figure it out