Let $X_1,...,X_n$ be IID random variable such that $X_i \sim N(0,1)$ for all $i=1,...,n$. Derive the distribution (i.e. give the pdf/cdf) of $\sum_{i=1}^{n} X_i^2$.
I know that from the definition of chi square the sum has chi square distribution. But what was the motivation for it?
What I've done so far is I found the distribution of $X_1^2$. Since $P(|X_1|<t)=2 \Phi(t) -1$ for $t>0$ (and 0 for $t<0$) thus $P(X_1^2<t)=2 \Phi(\sqrt{t}) -1$. No idea how I can find $P(X_1^2+X_2^2<t)$.
Once I find the formula for $P(\sum_{i=1}^{n} X_i^2<t)$ only then I can name it as a CDF of chi square distribution. Please help with finding $P(X_1^2+X_2^2<t)$ or give me any better approach.