Let $Z_{1}$ and $Z_{2}$ be i.i.d with $Z_{1}\sim N(0,1)$ and $Z_{2}\sim N(0,1)$
Let $X_{1}=m_{1}Z_{1}$ and let $X_{2}=m_{2}Z_{1}+m_{3}Z_{2}$
where $m_{1},m_{2},m_{3}$ are constants.
Find the conditional distribution of $f(X_{2}|(X_{1}=a))$ without using the definition of the conditional pdf. (Note $a$ is a constant)
My question is that when I write $a=m_{1}Z_{1}$ and I re-arrange to get $(a/m_{1})=Z_{1}$, then the left hand side is a constant wheras the right hand side is a random variable. Is this not a contradiction? Also, $E(Z_{1})=E(a/m_{1})=a/m_{1}$. But we know $Z_{1}$ is a standard normal, so it's expectation is $0$.
Can someone please tell me how to resolve this?