Continuous law of total probability with CDFs

Question

In my setup, each continuous random variable $X_i$ depends on $X_{i-1}$. I want to find $\Pr(X_i \leq x)$; For this, I use the law of total probability, conditioning on $X_{i-1}$. The law of total probability in the continuous case is (wikipedia)
$$
\Pr(A) = \int_{-\infty}^\infty \Pr(A \mid X = x) f_X(x) \ dx
$$
From this could I conclude the following?
$$
\Pr(X_i \leq x) = \int_{-\infty}^\infty \Pr(X_i \leq x \mid X_{i-1}=y) f_{X_{i-1}}(y) \ dy
$$
where
$$
f_{X_{i-1}}(y) = \frac{dPr(X_{i-1} \leq y)}{dy}
$$
(I put it this way to make it explicit that each $X_i$'s CDF can be found using the CDF of $X_{i-1}$)

Answer 1

Yes. If you do know the conditional cumulative distribution: $\mathsf P(X_i\leq x\mid X_{i-1}=y)$ and the probability density function: $f_{X_{i-1}}\!(y)$, for all supported $y$, then the Law of Total Probability says you may find the marginal cummulative distribution function $\Pr(X_i\leq x)$:

$$\Pr(X_i\leq x)=\int_{-\infty}^\infty \Pr(X_i\leq x\mid X_{i-1}=y)\,f_{X_{i-1}}\!(y)\,\mathrm d y$$