# Conditional probability of two dependent continuous random variables

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 $$p(V_1|V_2)$$?

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: $$p(V_1 \lvert V_2) = p(V_1, V_2) /p(V_2)$$

• Thank you very much! – Nadazhda Mar 23 at 10:37