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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

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Welcome to math.stackexchange!

This looks like a convolution problem. You can look up what convolutions are and how they are used (here's one resource), but it's also useful to show how one can obtain the main convolution formula from simple arguments since such arguments are not often shown.

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}

which is the standard formula for convolutions (https://en.wikipedia.org/wiki/Convolution#Definition).

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  • $\begingroup$ Thank you, this looks like what I'm looking for! When trying to Google the answer I kept stumbling on convolution stuff which I couldn't understand but the way you've presented it makes sense so thanks 🙂 $\endgroup$
    – Belisarius
    Commented Jan 12, 2021 at 8:30

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