Suppose, as in Exercise 52, that a stream of customers arrives at an $\mathrm{r}$-way junction, so that the time between successive arrivals has an exponential distribution with mean $1 / \lambda$. Then, by Theorem $3.2 .1(\mathrm{~g})$, the number of arrivals per unit of time has a Poisson distribution with mean $\lambda$. Suppose the branch selected by each arrival is chosen independently with the probability that an arrival takes path $i$ equal to $p_i$ for $i=1,2, \cdots, r$. We can imagine a random number generator that chooses 1 with probability $p_1, 2$ with probability $p_2, \ldots, r$ with probability $p_r$, where $\sum_{i=1}^r p_r=1$. Each customer then takes the path chosen by the random number generator and the generator makes a new choice for each arriving customer. Prove that the $i$ th output stream has a Poisson pattern with mean rate $p_i \lambda$. [Hint: Let $N(t)$ be the number of customer arrivals to the junction in $t$ time units. (We assume the observations begin when $t=0$.) Let $N_i(t)$ be the number of these arrivals that take the $i$ th path. Then the conditional joint distribution of $N_i(t)(i=1,2, \cdots, r)$ given that $N(t)=n$,
$$
P\left[N_1(t)=k_1, N_2(t)=k_2, \ldots, N_r(t)=k_r \mid N(t)=n\right],
$$
has a multinomial distribution (see Exercise 8 where event $E_i$ is the event that a customer takes path $i$ ). Multiplying this probability by the probability that $N(t)=n$, which has a Poisson distribution with mean $\lambda t$ by Theorem $3.2 .1(\mathrm{~g})$, we obtain the joint probability distribution $P\left(k_1, k_2, \ldots, k_r\right) . P\left(k_1, k_2, \ldots, k_r\right)$ expresses the probability that $k_1$ customers take the first path, $k_2$ take the second path, etc. Show that the joint probability factors into the product of $r$ Poisson probabilities.]