Chi-Squared Distribution from First Principles

Let $Z_1, \ldots, Z_n$ be i.i.d. $N(0,1)$ random variables and define $Q = \sum_{i=1}^n Z_i^2$.

(a) Derive the PDF of $Z_1^2$ using the change-of-variables technique for transformations of continuous random variables.

(b) Show that $Z_1^2 \sim \text{Gamma}\left(\frac{1}{2}, \frac{1}{2}\right)$ by matching the PDF from (a) to the Gamma family.

(c) Using the fact that the sum of independent Gamma random variables with the same rate parameter is Gamma, state the distribution of $Q$.

(d) Derive $E[Q]$ and $\text{Var}(Q)$.