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Let $X$ and $Y$ be two independent random variables that follow normal distributions with zero mean and variance equal to $\frac{\sigma^2}{2}$.

let $Z=X^2+Y^2$. I am looking for the distribution of a scaled version of $Z$, i.e. $Z_1=aZ$.

Could you please tell me how to compute the CDF or PDF of $Z_1$ ?

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    Hint: Use the fact that the sum of squares of N(0,1) variables is chi-square distributed. – 2017-02-22

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The sum of squares of two standard normals is chi-squared distributed on two degrees of freedom. This has the same PDF as an exponential with mean 2: $$f_{\chi^2(2)}(x) = \frac{1}{2}e^{-\frac{x}{2}} $$

What you have is a version of this, scaled by $a\frac{\sigma^2}{2}.$ So it's an exponential with mean $a\sigma^2,$ which has PDF $$ f_{aZ}(x) =\frac{1}{a\sigma^2}e^{-\frac{x}{a\sigma^2}}.$$

In general, for rescaled random variables, if the original PDF for $Z$ is $f_Z(x)$ the PDF for $aZ$ is $$f_{aZ}(x) = \frac{1}{a}f_Z(x/a).$$ This is a special case of the change of variables formula.