91Ó°ÊÓ

The special case of the gamma distribution in which \(\alpha\) is a positive integer \(n\) is called an Erlang distribution. If we replace \(\beta\) by \(1 / \lambda\) in Expression (4.8), the Erlang pdf is $$ f(x, \lambda, n)=\left\\{\begin{array}{cl} \frac{\lambda(\lambda x)^{x-1} e^{-\lambda x}}{(n-1) !} & x \geq 0 \\ 0 & x<0 \end{array}\right. $$ It can be shown that if the times between successive events are independent, each with an exponential distribution with parameter \(\lambda\), then the total time \(X\) that elapses before all of the next \(n\) events occur has pdf \(f(x ; \lambda, n)\). a. What is the expected value of \(X\) ? If the time (in minutes) between arrivals of successive customers is exponentially distributed with \(\lambda=.5\), how much time can be expected to elapse before the tenth customer arrives? b. If customer interarrival time is exponentially distributed with \(\lambda=.5\), what is the probability that the tenth customer (after the one who has just arrived) will arrive within the next 30 min? c. The event \(\\{X \leq t\\}\) occurs iff at least \(n\) events occur in the next \(t\) units of time. Use the fact that the number of events occurring in an interval of length \(t\) has a Poisson distribution with parameter \(\lambda t\) to write an expression (involving Poisson probabilities) for the Erlang cdf \(F(t, \lambda, n)=P(X \leq t)\).

Step by step solution

01

Determine the Expected Value of an Erlang Distribution

The expected value of an Erlang distribution is given by \( E(X) = \frac{n}{\lambda} \), where \( n \) is the shape parameter (number of events), and \( \lambda \) is the rate parameter. For part (a), with \( n = 10 \) and \( \lambda = 0.5 \), the expected value becomes \( E(X) = \frac{10}{0.5} = 20 \) minutes.

02

Determine the Probability Using Erlang Distribution

To find the probability that the tenth customer arrives within 30 minutes, we need to calculate \( P(X \leq 30) \). The cumulative distribution function (CDF) of the Erlang distribution is given by the sum of Poisson probabilities: \[ P(X \leq t) = \sum_{k=n}^{\infty} \frac{(\lambda t)^k e^{-\lambda t}}{k!} \]. For \( n = 10 \), \( \lambda = 0.5 \), and \( t = 30 \), calculate the sum of Poisson probabilities from \( k = 10 \) to \( \infty \) with rate parameter \( \lambda t = 15 \).

03

Express Erlang Cumulative Distribution with Poisson Probabilities

The Erlang cumulative distribution function (CDF) is related to Poisson distribution probabilities for \( k \geq n \) as explained. For any \( t \), the Erlang CDF is \( F(t, \lambda, n) = 1 - \sum_{k=0}^{n-1} \frac{(\lambda t)^k e^{-\lambda t}}{k!} \). This uses the complement rule for Poisson probabilities where fewer than \( n \) events happen within time \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gamma Distribution

The gamma distribution is a continuous probability distribution that is highly useful in various fields like engineering, science, and statistics. It characterizes the time needed for a certain number of events to occur in a Poisson process. This makes it a generalization of the Erlang distribution, which is a special case of the gamma distribution.The key parameters of the gamma distribution are:

• Shape parameter (\( \alpha \)): This is often denoted by "k" or "n" and dictates the shape of the distribution.
• Rate parameter (\( \beta \) or \( \lambda \)): This inverse value of the scale parameter controls the rate of events occurring.

The probability density function (pdf) of the gamma distribution can be expressed as:\[f(x; \alpha, \lambda) = \frac{ \lambda^{\alpha} x^{\alpha-1} e^{-\lambda x} }{ (\alpha - 1)! }\]The gamma distribution is flexible and can be used to model skewed distributions. When \( \alpha \) is an integer, the distribution is known as the Erlang distribution.

Expected Value

The expected value, often denoted as \( E(X) \), represents the average or mean value for a set of possible outcomes, weighted according to their probabilities. For the Erlang distribution, which is a specific form of the gamma distribution, the expected value is given by the formula:\[E(X) = \frac{n}{\lambda}\]Here:

• \(n\) is the shape parameter, indicating the number of events or trials.
• \(\lambda\) is the rate parameter, representing how often events occur.

In practical terms, if you consider something like the arrival of customers, the expected time between successive customer arrivals can easily be calculated using this formula. For example, if \(n \) is 10 and \(\lambda = 0.5\), the expected time before all events occur is 20 minutes. This concept helps in predicting average outcomes and planning accordingly.

Poisson Distribution

The Poisson distribution is instrumental when analyzing the frequency of events in a fixed period of time or space. It is often used when events are scattered sparsely over a period or region. The Poisson distribution can also represent the number of events occurring in an interval if these events happen with a constant mean rate and independently of time since the last event.The primary characteristic of the Poisson distribution is its single parameter \( \lambda \), which signifies the average number of occurrences in the interval. The probability mass function for the Poisson distribution is given by:\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\]for \( k = 0, 1, 2, \, ... \)In the context of the exercise, a Poisson distribution is leveraged to model situations where specific event occurrences, like customers arriving, are analyzed over time. Particularly, the Erlang distribution uses the Poisson probabilities to calculate its own metrics like the CDF (cumulative distribution function). This distribution is directly linked to the Erlang distribution as they both stem from dealing with random processes of events.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time until an event occurs. It is particularly applicable when dealing with processes where events happen continuously and independently at a constant average rate. The defining feature of the exponential distribution is its parameter, \( \lambda \), representative of the rate of the event happening. The probability density function (pdf) is expressed as:\[f(x; \lambda) = \lambda e^{-\lambda x}\]where \( x \geq 0 \).In scenarios where events occur one after another at a steady rate, but without memory of past events, such as arrivals of customers, this distribution excellently models the time between successive events. The exponential distribution is significant because of its relationship with the Poisson process, where it models time between independent events. This is also why it connects directly to the gamma and Erlang distributions, serving foundational purposes in calculating times for multiple events in the Erlang framework.