I know that if a random variable is defined like this,
$\Delta_{N}=\sum_{i=1}^{N}Z_{i}^{2}$
where, $Z_{i}\sim N(0,1)$. Then $\Delta_{N}$ is called chi-squared random variable of N degree(s) of freedom.
Now my question is, let us define a new random variable,
$\Delta_{N}^{'}=\sum_{i=1}^{N}\alpha_{i}Z_{i}^{2}$
where $\alpha_{i}$'s are constant coefficients.
Question is what type of distribution they will follow?
On
Since definition of chi-square is been concerted through that definition, hence $\Delta_{N}^{'}$ will not obey chi-square distribution. That is technically correct answer according to me. But on the same time since $\alpha_{i}$'s are constant coefficients, hence qualitative characteristics of density functions of $\Delta_{N}^{'}$ will be same as chi-square density function as DOF changes. Have a look at density functions of `pure' chi-square RV -
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Reason why I think so is that, if someone questions that why for DOF=1 density function peaks at 0? Answer is, if apply transformation formula of PDF for normal distribution to get density for chi-squared then you will get a factor like this, $\frac{e^{-\frac{x^{2}}{2}}}{x}$. And hence infinite peak at zero. Now this argument will remain unchanged whether we have a extra constant coefficient $\alpha_{i}$ in definition. Hence I have said that `qualitative characteristics will remain same'. Similar arguments can be raised for DOFs>1.
Looking forward for any comments on my answer. Thank you in advance.
It will be a Gamma distribution.
By looking at the moment generating function of $Y_i=\alpha_iZ_i^2$, you will see that
$$ M_{Y_i}(t) = (1-2\alpha_it)^{\frac{N}{2}} $$
Which demonstrates that $Y_i\sim Gamma(2\alpha_i,\frac{N}{2})$. We know that the sum of Gamma distributions with the same second parameter is itself a Gamma where the first parameter of the sum is the sum of the first parameters of the individuals.
$$ \Delta'_N\sim Gamma(\sum_{i=1}^N2\alpha_i,\frac{N}{2}) $$