# Exponential distribution examples and solutions pdf

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- 5.4: The Exponential Distribution
- Exponential Distribution — Intuition, Derivation, and Applications
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*The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time beginning now until an earthquake occurs has an exponential distribution.*

The Exponential Distribution is another important distribution and is typically used to model times between events or arrivals. It is assumed that the average time customers spends on hold when contacting a gas company's call centre is five minutes. If the company employs a new team, at some expense,then the average waiting time is reduced to four minutes.

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Students arrive at a local bar and restaurant according to an approximate Poisson process at a mean rate of 30 students per hour. What is the probability that the bouncer has to wait more than 3 minutes to card the next student? Now, we just need to find the area under the curve, and greater than 3, to find the desired probability:. The number of miles that a particular car can run before its battery wears out is exponentially distributed with an average of 10, miles.

The owner of the car needs to take a mile trip. What is the probability that he will be able to complete the trip without having to replace the car battery? At first glance, it might seem that a vital piece of information is missing. It seems that we should need to know how many miles the battery in question already has on it before we can answer the question!

Now, suppose the following is true:. It turns out that the above statement is true for the exponential distribution you will be asked to prove it for homework!

It is for this reason that we say that the exponential distribution is " memoryless. It can also be shown do you want to show that one too? We'll leave it to the gentleman in question to decide whether that probability is large enough to give him comfort that he won't be stranded somewhere along a remote desert highway! Breadcrumb Home 15 Font size. Font family A A. Content Preview Arcu felis bibendum ut tristique et egestas quis: Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris Duis aute irure dolor in reprehenderit in voluptate Excepteur sint occaecat cupidatat non proident.

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Excepturi aliquam in iure, repellat, fugiat illum voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos a dignissimos. Close Save changes. Help F1 or? Solution At first glance, it might seem that a vital piece of information is missing. Save changes Close.

## 5.4: The Exponential Distribution

Let X be a random variable with moment generating function MX. For simplicity, here we use the PDF as an illustration. For example, the amount of time beginning now until an earthquake occurs has an exponential distribution. Similarly, the probability that the next thing to happen is the next 9. It is also known as Cauchy-Lorentz distribution, the Cauchy distribution, or the Breit-Wigner and geometric distributions are all exponential families.

Sign in. To predict the amount of waiting time until the next event i. For example, we want to predict the following:. Does the parameter 0. For example, your blog has visitors a day. That is a rate.

For example, the amount of time beginning now until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution. Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

## Exponential Distribution — Intuition, Derivation, and Applications

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time beginning now until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution. Values for an exponential random variable occur in the following way.

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time beginning now until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

*Students arrive at a local bar and restaurant according to an approximate Poisson process at a mean rate of 30 students per hour.*

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Documentation Help Center. Compute the density of the observed value 5 in the standard exponential distribution. Compute the density of the observed value 5 in the exponential distributions specified by means 1 through 5. Compute the density of the observed values 1 through 5 in the exponential distributions specified by means 1 through 5 , respectively.

Documentation Help Center. The exponential distribution is a one-parameter family of curves. The exponential distribution models wait times when the probability of waiting an additional period of time is independent of how long you have already waited. For example, the probability that a light bulb will burn out in its next minute of use is relatively independent of how many minutes it has already burned.

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