Is it true that multiplication of a chi-square random variable by a real constant remains chi-square? I tried to check this using a change of variables, but it didn't look promising.
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kjetil b halvorsen
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half-pass
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It has the same form as chisquared, and is usually called scaled chisquare. – kjetil b halvorsen Oct 07 '15 at 15:06
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@half-pass A scaled chi-square is not chi-square if the scaling factor is anything but 1. It is still gamma though. – Glen_b Oct 07 '15 at 15:16
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Multiplication by a constant changes the scale parameter of a gamma distribution. Since a chi-squared distribution is a special case of a gamma distribution with scale equal to $2$, it is easy to see that if you multiply the random variable with a constant it no longer follows the chi-squared distribution.
JohnK
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Well, I suspect this is self-study, here are some hints:
- The characteristic function of a $\chi^2$ is $\phi(t)=(1-2it)^{-k/2}$
- The characteristic function of a $\Gamma(k,\theta)$ is $\phi(t)=(1-\theta it)^{-k}$
(where I worked with wikipedia's notations).
What can you say about $\phi_{cX}$ and $\phi_X$ ?
RUser4512
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It is not true that multiplication of a chi-square random variable by a real constant remains chi-square.
Chi-square is sum of square of independent standard normal distribution. Multiplying a non-one constant will make the summation not by standard normal anymore.
Tan
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Yes, therefore it should be called "scaled chi-square", as kjetil b halvorsen commented :-) – Ute Oct 01 '23 at 11:27