what is the formal notation about the CDF or PDF of a function with limited interval.

Given an arbitrary function, like $d(t)$, within some finite interval $[t_1,t_2]$, we can get its PDF and CDF by sampling, as shown like

My question is where there is any formal notation to describe them. The reason why I do not call them CDF and PDF is because $d(t)$ is deterministic.

My final aim is to use some formal notation to describe the following question $$ \int_{t_1}^{t_2}g\left(d\left(t\right)\right)\cdot \mathbb{I}\left\{b(t)<\bar{B}\right\} \ dt=(t_2-t_1)\int_{0}^{\bar{B}}g\left(u\right)\ dF_{d(t)}(u) $$ where $F_{d(t)}(u)$ is the CDF of $d(t)$ within $[t_1,t_2]$.

Solution 1:

Your plots approximate the CDF and PDF of the r.v. $Y=d(T),$ where r.v. $T\sim\text{Uniform}[t_1,t_2].$ Your equation should therefore be $$\begin{align}\int_{t_1}^{t_2}g\left(d\left(t\right)\right)\cdot \mathbb{I}\left\{b(t)<\bar{B}\right\} \ dt &=(t_2-t_1)\,\mathbb{E}\left[g\left(d\left(T\right)\right)\cdot \mathbb{I}\left\{b(T)<\bar{B}\right\} \right]\\ &=(t_2-t_1)\,\mathbb{E}\left[g\left(Y\right)\cdot \mathbb{I}\left\{Y\in C\right\} \right]\\ &=(t_2-t_1)\int_{C}g\left(y\right)\ dF_{Y}(y)\end{align}$$ where $C=\{d(t):\ b(t)<\bar B,\ t\in[t_1,t_2] \}.$ The set $C$ is determined by both functions $d(.)$ and $b(.),$ and might not be $[0,\bar B]$ as you've written.