Two restaurants, and , receive orders independently according to Poisson processes with rates of 4 and 2 orders per minute, respectively.
Let be the total number of orders received by both restaurants in a period of minutes.
Derive the moment generating function for the total number of orders .
[4]Orders arrive at restaurant X according to a Poisson process with a mean of 4 per 15 minutes. Orders arrive at restaurant Y according to a Poisson process with a mean of 2 per 15 minutes. The arrivals at the two restaurants are independent.
Exactly 25 orders arrived at the two restaurants in a 15-minute period. Calculate the probability that none of these orders came from restaurant Y.
[3]Two restaurants, X and Y, receive online orders according to independent Poisson processes. Restaurant X receives orders at a mean rate of per minute, and restaurant Y receives orders at a mean rate of per minute.
Given that in a -minute interval the two restaurants together receive orders, calculate the probability that exactly of those came from restaurant X.
[4]Two restaurants, X and Y, receive orders according to independent Poisson processes with rates per minutes and per minutes respectively. Let and be the number of orders received by restaurant X and restaurant Y respectively in a -minute period.
Given that in a particular -minute period the two restaurants together receive orders, calculate the conditional expectation and the conditional variance .
[4]Orders arrive at Restaurant X and Restaurant Y independently according to Poisson processes. Restaurant X receives an average of 4 orders per minute, and Restaurant Y receives an average of 2 orders per minute.
Calculate the probability that the very first order (across both restaurants) in a given minute comes from Restaurant X.
[2]Restaurant X receives orders as a Poisson process at a rate of orders per minute and restaurant Y at a rate of orders per minute. Assuming the processes are independent, calculate the probability that in a -minute period they receive exactly orders in total.
[3]Two restaurants receive orders independently such that the number of orders follows a Poisson process. The combined rate of orders for the two restaurants is orders per minute.
Approximate the probability that the two restaurants together receive more than orders in a -minute period, using a normal approximation.
[5]Orders arrive at a restaurant from two independent sources. Orders from the first source arrive at a mean rate of per minute, and orders from the second source arrive at a mean rate of per minute. The arrivals from each source follow a Poisson distribution.
Calculate the probability that the restaurant receives more than orders in a -minute period.
[3]Two independent restaurants, and , receive online orders according to Poisson processes. Restaurant receives orders at a mean rate of 4 orders per minute, and restaurant receives orders at a mean rate of 2 orders per minute.
Find the probability that, in a 6-minute period, at least one of the two restaurants receives no orders.
[4]Orders arrive at a store from two independent sources. Orders from source A arrive according to a Poisson process with a rate of 4 per minute. Orders from source B arrive according to an independent Poisson process with a rate of 2 per minute. Let be the time, in minutes, until the 10th total order arrives at the store from either source.
Find the distribution of , stating the name of the distribution, its parameters, and its probability density function.
[5]Two restaurants, X and Y, receive orders independently. Restaurant X receives orders at an average rate of 4 per minute, and restaurant Y receives orders at an average rate of 2 per minute.
Find the probability that in a 5-minute period, restaurant X receives exactly 15 orders and restaurant Y receives exactly 10 orders. Give your answer to three significant figures.
[3]Three restaurants X, Y, Z receive orders as independent Poisson processes at rates , , and per minute respectively. What is the probability that in minutes the total number of orders exceeds ?
[4]