Fred lives in Blissville, where buses always arrive exactly on time, with the time between successive buses fixed at $10$ minutes. Having lost his watch, he arrives at the bus stop at a uniformly random time on a certain day (assume that buses run $24$ hours a day, every day and that the time that Fred arrives is independent of the bus arrival process).
Question: What is the average time and the PDF that Fred has to wait?
I find that the average time is in the range $[0,10]$ so the mean is $5$ minutes but I couldn't find the PDF.
Let's start by letting $T$ represent the amount of time, in minutes, Fred has to wait at the bus stop. One thing we know right away is $T\ge0$, since Fred can't possibly wait a negative amount of time at the stop. We also know that the bus arrives every 10 minutes on a fixed schedule, so the longest he could possibly wait is 10 minutes. This tells us $T\le 10$. So far, we have:
$$ P[T < 0] = 0 , $$ $$ P[T > 10] = 0 . $$
Now, in that range of $ 0 \le T \le 10 $, we also know that Fred's arrival is uniformly random and independent of the bus's arrival. This means that all possible wait times between $T = 0$ and $T = 10$ are equally likely - this is how we know Fred's wait time is uniformly distributed:
$$ T \sim \mathcal{U}(0,10) $$
Broadly, for a random variable $ X \sim \mathcal{U}(a,b) $, the density function is given by
$$ f(x) = \begin{cases}\frac{1}{b-a} & x\in[a,b]\\ 0 & x\notin[a,b] \\ \end{cases} .$$
The reason this is the functional form we want is it's a constant value across the domain, which is what we're looking for when we say that any time between 0 and 10 is "equally likely."
Note, for a continuous distribution, $P[X=x]=0$, so we're not saying that the probability of waiting 1 minute is equal to the probability of waiting 2 minutes - both of those values are 0 - it's meant to be a general description of what the density looks like - a horizontal line.
In this case, $a=0$ and $b=10$, so we have $b-a=10$ and
$$ f(t) = \begin{cases}\frac{1}{10} & t\in[0,10]\\ 0 & t\notin[0,10] \\ \end{cases} .$$