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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?

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

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

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  • $\begingroup$ Thanks. I'm struggling to map this with my question. When $\mathcal{T}$ is not finite, is it $$F_{Y,V}(y,v)=F_{Y|V}(y|\{i\in \mathcal{V}: i\leq v\}) \times F_V(v)= \int_{\mathcal{T}} F_{Y|V,T}(y|\{i\in \mathcal{V}: i\leq v\}, t)dF_{T|V}(t|\{i\in \mathcal{V}: i\leq v\}) \times F_V(v)$$? $\endgroup$ – user3285148 Jun 27 at 13:22
  • $\begingroup$ But what if $\mathcal{T}$ is finite? $\endgroup$ – user3285148 Jun 27 at 13:23
  • $\begingroup$ 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 $Y$. $\endgroup$ – Ben Jun 27 at 13:27
  • $\begingroup$ 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 $V$. $\endgroup$ – user3285148 Jun 27 at 13:29

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