Question Type 4: Solving word problems involving multiple poisson distributions Exercises

Restaurant X receives orders as a Poisson process at rate $4$ orders per minute and restaurant Y at rate $2$ orders per minute. What is the probability that in $5$ minutes they receive exactly $35$ orders in total?

Using the same rates ($4$ and $2$ per minute), calculate the probability that they receive more than $70$ orders in $10$ minutes.

What is the probability that in $5$ minutes restaurant X receives exactly $15$ orders and restaurant Y receives exactly $10$ orders?

What is the probability that the very first order (across both restaurants) in a given minute comes from restaurant X?

What is the probability that in a $6$-minute period at least one of the two restaurants receives no orders?

Three restaurants X, Y, Z receive orders as independent Poisson processes at rates $4$, $2$, and $3$ per minute respectively. What is the probability that in $10$ minutes the total number of orders exceeds $80$?

Given that in a $4$-minute interval the two restaurants together receive $20$ orders, what is the probability that exactly $14$ of those came from X?

Given that exactly $25$ orders arrived in $15$ minutes, what is the probability that none came from restaurant Y?

Derive the moment generating function $M_N(s)$ of the total number of orders $N$ received by both restaurants in $t$ minutes.

In $10$ minutes the two restaurants together receive $30$ orders. Find the conditional expectation $E[N_X\mid N_X+N_Y=30]$ and conditional variance $\mathrm{Var}(N_X\mid N_X+N_Y=30)$, where $N_X$ and $N_Y$ are counts for X and Y.

Find the distribution of the time $T_{10}$ until the $10$th combined order arrives.

Approximate the probability that the two restaurants together receive more than $100$ orders in $20$ minutes using a normal approximation.