# Modifying known probability density function using another pdf as an initial condition

My question is for a brownian motion type process with two physically different regions. Region 1 is between points A and B and region 2 is between points B and C. The time taken to travel between two points A and B is a stochastic quantity $$T_1$$. Say I know that the time taken to travel between A and B follows a gaussian distribution i.e.

$$P(T_1) = \frac{1}{\sigma \sqrt{2 \pi}} \exp\bigg[ -\frac{1}{2} \frac{(T_1 - \langle T_1\rangle)^2}{\sigma^2} \bigg]$$

where $$\sigma$$ is the standard deviation of the distribution as expected. And that I know once you enter region 2 the time to travel between the points B and C is another stochastic quantity $$T_2$$ with its own pdf $$G(T_2)$$ that is generically not gaussian. As an example

$$G(T_2) = \alpha \cdot T_2 \cdot e^{-\beta\cdot T_2^2}$$

but in practice it could be any kind of pdf.

My question is how can you combine the knowledge of the pdf for $$T_1$$ and seperately the pdf for $$T_2$$ to work out the pdf for the time taken to travel the total distance from A to B and then to C, $$T_3$$. Intuitively given that $$P(T_1)$$ is a simple gaussian distribution. I would expect the full pdf to just be a more spread out version of $$G(T_2)$$ but I am not sure how to show this explicitly or quantify it.

Any help to how I can combine pdfs in this way (even if it has to be numerically) would be greatly appreciated

Let $$P(t_1)$$ be the probability density for moving from $$A$$ to $$B$$ in time $$t_1$$ and $$G(t_2)$$ be the probability density for moving from $$B$$ to $$C$$ in time $$t_2$$. Then, let's denote the probability density for moving from $$A$$ to $$B$$ to $$C$$ as $$H(t_\text{tot})$$. Noting that $$t_{\text{tot}} = t_1 + t_2$$ by constraint, we can find the probability density for $$H$$ by integrating our distributions of $$P(t_1)$$ and $$G(t_2)$$ over their domains subject to that constraint. We then have
\begin{align} H(t_\text{tot}) &= \int^{\infty}_{-\infty} dt_1 \int^{\infty}_{-\infty} dt_2\, \delta(t_\text{tot}- t_1 - t_2) P(t_1) G(t_2)\\[.5em] & = \int^{\infty}_{-\infty} dt_1\, P(t_1) G(t_{\text{tot}}-t_1) \end{align}