# Application of law of total probability for continuous random variables

Consider 3 random variables $$Y,V,T$$, with supports $$\mathcal{Y},\mathcal{V},\mathcal{T}$$, respectively.

Let

• $$P_{Y,V}$$ denote the probability distribution of $$(Y,V)$$

• $$P_{V}$$ denote the probability distribution of $$V$$

• $$P_{T|v}$$ denote the probability distribution of $$T$$ conditional on $$V=v$$

• $$P_{Y|v,t}$$ denote the probability distribution of $$T$$ conditional on $$V=v, T=t$$

Suppose that all the supports are finite sets. Then, we know that, by the law of total probability:

$$P_{Y,V}(y,v)=P_{V}(v)\sum_{t\in \mathcal{T}}P_{T|v}(t) P_{Y|v,t}(y)$$

Now, I want to rewrite the same expression when $$V$$ is a continuous random variable. I don't want to introduce densities. If necessary, I can work with cumulative distribution functions. Could you help to provide a notationally precise statement?

The law of total probability can be written in terms of the CDFs as a Riemann-Stieltjes integral:

$$F_X(x) = \int \limits_\mathscr{Y} F_{X|Y}(x|y) \ dF_Y(y).$$

• Thanks. I'm struggling to map this with my question. When $\mathcal{T}$ is not finite, is it $$F_{Y,V}(y,v)=F_{Y|V}(y|\{i\in \mathcal{V}: i\leq v\}) \times F_V(v)= \int_{\mathcal{T}} F_{Y|V,T}(y|\{i\in \mathcal{V}: i\leq v\}, t)dF_{T|V}(t|\{i\in \mathcal{V}: i\leq v\}) \times F_V(v)$$?
– Star
Commented Jun 27, 2019 at 13:22
• But what if $\mathcal{T}$ is finite?
– Star
Commented Jun 27, 2019 at 13:23
• The conditional CDF in my answer is still conditional on a single point, so $F_{X|Y}(x|y) \equiv \mathbb{P}(X \leqslant x | Y=y)$. In the Riemann-Stieltjes integral, if $Y$ is countable (including finite) then the integral reduces to a sum taken over the mass function of $Y$.
– Ben
Commented Jun 27, 2019 at 13:27
• But I think that the CDF conditional on a single point is not what I need in my specific case, because everything start from the joint of $Y$ and $V$.
– Star
Commented Jun 27, 2019 at 13:29