I have two continuous random variables $V_1$ and $V_2$ defined as
$$\begin{aligned}V_1 &:= a_1 \cdot W_1 + a_2 \cdot W_2 + a_3 \cdot W_3 + a_4 \cdot W_4 + a_5 \cdot W_5 \\ V_2 &:= b_1 \cdot Y + b_2 \cdot W_2 + b_3 \cdot W_3 + b_4 \cdot W_4 + b_5 \cdot W_5\end{aligned}$$
where $W_1$, $W_2$, $W_3$, $W_4$, $W_5$ and $Y$ are mutually independent continuous random variables with known Gaussian distributions. Could anyone please help me with the methodology of finding a conditional probability density function $p(V_1|V_2)$?
Best Answer
Let $\{W_i\}$ be set of $W_i$ for $i=2,3,4,5$.
$V_1$ and $V_2$ are not independent. But there are conditionally independent given $\{W_i\}$!. Therefore:
$$p(V_1, V_2 \lvert \{W_i\}) = p(V_1 \lvert \{W_i\}) p(V_2 \lvert \{W_i\})$$
From law of total probability:
$$p(V_1, V_2 ) = \int p(V_1, V_2 \lvert \{W_i\}) p( \{ W_i \} )$$
Then you can easily compute $p(V_2)$ since $\{W_i\}$ are mutually independent Gaussians, so sum of them is another Gaussian, $\sum W_i = N(\sum \mu_i, \sum \sigma^2_i)$, where $\mu_i, \sigma^2_i$ are means and variances of $W_i$.
Then, from definition: $$p(V_1 \lvert V_2) = p(V_1, V_2) /p(V_2)$$