Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$X_1$ and $X_2$ are independent random variables. $X_1$ and $Y = X_1 + X_2$ have $\chi^2$ distributions with $r_1$ and $r$ degrees of freedom and $r_1 < r$.

How would I show that $X_2$ has a $\chi^2$ distribution with $r - r_1$ degrees of freedom?

share|improve this question

1 Answer 1

up vote 4 down vote accepted

Since $X_1$ and $X_2$ are independent, the characteristic function of $Y$ is the product of the characteristic functions of $X_1$ and $X_2$, the characteristic function of a $\chi^2$ distribution with $r$ degrees of freedom is $(1-2ik)^{-r/2}$ ($k$ is theFourier variable). Thus the characteristic function of $X_2$ is $(1-2ik)^{-(r-r_1)/2}$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.