Assume that the traffic, one afternoon at a certain point in time, both in north and southbound, follows a Possion process with intensities $7$ veichles per minute southbound and $3$ veichles per minute northbound.
a) What is the probability that under a given minute, a total of exactly $7$ cars pass?
b) At a certain moment, what is the probability that the next veichle that passes is southbound?
c) Given that a total of exactly $7$ veichles have passed during one minute, what is the conditional CDF for the number of veichles that are southbound?
Solution:
a): Let the $X,Y$ be the number of veicheles that go south and north respectively. Then $X\sim\text{Poi}(7), \ Y\sim\text{Poi}(3)$ and by independence $X+Y\sim\text{Poi}(10).$ So,
$$P(X+Y=7)=\frac{e^{-10}10^7}{7!}=0.09.$$
b): Let the time until the next car is seen going south be $T_s\sim\text{exp}(7)$ and let the time until the next car seen is going north be $T_n\sim\text{exp}(3)$. Then we are looking for
$$P(T_s<T_n)=\int_0^{\infty}P(T_n>x)f_{T_s}(x) \ dx = \int_0^{\infty}(1-P(T_n<x))f_{T_s}(x) \ dx = $$
$$=7\int_0^{\infty}e^{-10x} \ dx=\frac{7}{10.}$$
c) It follows that
$$P(X=x|X+Y=7)=P(X=x|Y=7-x)=\frac{P(X=x,Y=7-x)}{P(X+Y=7)}=$$
$$=\frac{P(X=x)P(Y=7-x)}{P(X+Y=7)}=\frac{\frac{e^{-7}7^x}{x!}\cdot\frac{e^{-3}3^{7-x}}{(7-x)!}}{\frac{e^{-10}10^7}{7!}}={7\choose x}\left(\frac{7}{10}\right)^x\left(\frac{3}{10}\right)^{7-x}.$$
Questions:
1) In b), first equation line. When $P(T_s<T_n)$ is computed, I don't understand what is happening there. I assume that the law of total proability is involved in the form
$$P(Y\in B)=\int_{-\infty}^{\infty}P(Y\in B|X=x)fX(x) \ dx. \tag1$$
However I can't make sence of (1) to my problem. The notations are confusing me. In what way for example, is $P(T_s<T_n)$ similar to $P(Y\in B)?$
2) In c), the computations seem to be taken out of the thin air. For example, why is this wrong:
$$P(X=x|X+Y=7)=P(X=x|Y=7-x)=\frac{P(X=x,Y=7-x)}{P(Y=7-x)}=P(X=x),$$
by independence. Why can't I also change the $P(X+Y=7)$ to $P(Y=7-x)$ in the denominator like I did in the numerator?