Customers arrive at a bank at a Poisson rate λ. Suppose two...

Customers arrive at a bank at a Poisson rate $\lambda$. Suppose two customers arrived during the first hour. What is the probability that
(a) both arrived during the first 20 minutes?
(b) at least one arrived during the first 20 minutes?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Poisson Process

A Poisson process models random events occurring independently over time at a constant average rate. It is characterized by the property that the number of events in disjoint intervals are independent, and the probability of one event occurring in a very short interval is proportional to the length of the interval. This framework is widely used to model arrival processes, such as customers arriving at a bank.

Conditional Uniform Distribution of Arrival Times

Given a fixed number of arrivals in a Poisson process over a time interval, the individual arrival times are independently and uniformly distributed over that interval. This means that if we know exactly two arrivals have occurred in one hour, the times at which these arrivals occur can be considered as two independent random draws from a uniform distribution spanning that hour.

Binomial Splitting Property

Using the fact that the arrival times are conditionally uniform, the number of arrivals in any subinterval follows a binomial distribution. The probability of any single arrival falling within a particular subinterval is the length of the subinterval divided by the total interval length. This property allows the calculation of probabilities such as both arrivals occurring in the first 20 minutes or at least one arrival occurring in that period.

Transcript

Please give Ace some feedback