From my window in the office, I can see a bus station, where buses come in average frequency of 15 minutes and the time between arrival of two buses distributed exponentially. One day, I went out my office to the station 10 minutes after a bus arrived. Suppose I walk 6 minutes:
a.What the probability I'll miss a bus?
b. What's the probability I'll miss two buses with independent arrival times?
c.What's the distribution of my waiting time in the station (after I arrived)?
About A+B: since the distribution of time between arrivals distributed exponentially, In A I found $P(X<16)$ Where X is 1st bus arrived at the station. In B I found $P(X\le a,Y\le 16-a)$ and since the arrivals are independent, $=P(X\le a)\cdot P(Y\le16-a)$. We can find all of them but I wonder if I had to express the answer in B without terms which include $a$.
About C:How generally can I find distributions given any details?
I'll be glad for help with C.
EDIT: (A) we had to calculate $P(X<6)$ (from 0 since waiting time is non-negative) equals $=\int_0^6\frac {e^{-\frac x {15}}}
{15}=0.33$
. and in B :$P(X+Y<6)=\int_{0}^{6}\frac{e^{\frac{-t}{15}}}{15}\cdot \frac{e^{\frac{-6-t}{15}}}{15}dt=0.018$
About C: I'm still stuck…
Best Answer
Let $X$ denote the time (measured in minutes) between the last bus which arrived at the bus station before you left your office and the next one. Let $Y$ denote the time (measured in minutes) between the next bus and the next bus after it. By hypothesis, the couple $(X,Y)$ is i.i.d. exponential with known parameter $\lambda=1/15$.
Question (a) asks for $P[X\lt16\mid X\gt10]$. This is not $P[X\lt16]$ but $P[X\lt6]$. Do you see why? Can you compute this value?
Additional hint: If $Z$ is exponentially distributed, then $P[Z\gt x+y\mid Z\gt y]=$ $____$.
Question (b) asks for $P[X+Y\lt16\mid X\gt10]$. The answer is $P[X+Y\lt6]$. Do you see why? Can you compute this value?
Question (c) refers to the same quite general property of Poisson processes which was used in questions (a) and (b). Do you see this property? Can you deduce the answer?