Prove that a random variable with density function

\[ p(x)=\left\{\begin{array}{cc} 0 & \text { for } x \leqslant 0 \\ \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e_{e-\beta x} & \text { for } x>0 \end{array}\right. \]

where \(\alpha>0, \beta>0\) are constants, is infinitely divisible.

From this it follows, in particular, that the Maxwell distribution and the chi-square distribution are infinitely divisible for any value of \(n\).