When αis a positive integer n, the gamma distribution is...

When $\alpha$ is a positive integer $n$, the gamma distribution is also known as the Erlang distribution. Setting $\alpha=n$ in the gamma distribution on page 215 , the Erlang distribution is $$f(x)=\left\{\begin{array}{ll}\frac{x^{n-1} e^{-x / \beta}}{\beta^{n}(n-1) !}, & x>0 \\0, & \text { elsewhere }\end{array}\right.$$ It can be shown that if the times between successive events are independent, each having an exponential distribution with parameter $\beta,$ then the total elapsed waiting time $X$ until all $n$ events occur has the Erlang distribution. Referring to Review Exercise $6.62,$ what is the probability that the next 3 calls will be received within the next 30 minutes?

When $\alpha$ is a positive integer $n$, the gamma distribution is also known as the Erlang distribution. Setting $\alpha=n$ in the gamma distribution on page 215 , the Erlang distribution is
$$f(x)=\left\{\begin{array}{ll}\frac{x^{n-1} e^{-x / \beta}}{\beta^{n}(n-1) !}, & x>0 \\0, & \text { elsewhere }\end{array}\right.$$
It can be shown that if the times between successive events are independent, each having an exponential distribution with parameter $\beta,$ then the total elapsed waiting time $X$ until all $n$ events occur has the Erlang distribution. Referring to Review Exercise $6.62,$ what is the probability that the next 3 calls will be received within the next 30 minutes?

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Key Concepts

Exponential Distribution

The exponential distribution describes the time between independent events occurring at a constant average rate. It has a memoryless property, meaning the chance of an event occurring in the future does not depend on how much time has already elapsed.

Gamma Distribution

The gamma distribution generalizes the exponential distribution by modeling the waiting time until multiple events occur. It is defined by a shape and a scale parameter, making it versatile for describing sums of independent exponentially distributed waiting times.

Erlang Distribution

The Erlang distribution is a special case of the gamma distribution where the shape parameter is a positive integer. It represents the total waiting time for a fixed number of events to occur, particularly in settings like call centers or other queuing systems.

Poisson Process

A Poisson process models events that occur randomly and independently over time with a constant average rate. It underpins the exponential and Erlang distributions, as the interarrival times follow an exponential distribution and their sum (waiting for several events) follows an Erlang distribution.

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