If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

Question

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note that it's distance between and so either car may be ahead of the other.) Find:

(i) P$(X<a)$ (ii)P$(X=a)$

for any real number $a\geq 0$.

Answer 1

Denote by $N_1\sim N(\mu_1,\sigma_1^2)$ and $N_2\sim N(\mu_2,\sigma_2^2)$ the respective speeds of cars 1 and 2.

I will assume in the following that $N_1$ and $N_2$ are independent, and that both cars start at the same point.

After $m$ hours, the first car will be at a distance $mN_1$ from the origin and the second car will be at a distance $mN_2$ from the origin. Note that $N_1$ or $N_2$ may be negative, i.e. the cars may be going in opposite directions.

Either way, the distance between both cars is $X=m\left|N_1-N_2\right|$.

Now, do you know what the distribution of $N_1-N_2$ is? If not, look at the characteristic function, and use the independence of $N_1$ and $N_2$.