Exponential Distribution

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The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, which is a process that occurs at a constant rate. It is commonly used in modelling waiting times, such as the time between arrivals at a service counter or the time between machine failures. The exponential distribution is defined by a single parameter, \(\lambda\), which is the rate at which the events occur.

Properties of Exponential Distribution:

The exponential distribution has several important properties, including:

  • Memoryless property: The exponential distribution is characterized by the fact that the waiting time until the next event is independent of the time that has already passed since the last event. This is known as the memoryless property.
  • Right-skewed shape: The exponential distribution is right-skewed, meaning that it has a long tail on the right side of the distribution.
  • Non-negative: The exponential distribution is always non-negative since the time between events can never be negative.
  • Asymptotic behaviour: The exponential distribution approaches 0 as the values of x become increasingly large. This means that the probability of a value being extremely large is always non-zero but becomes increasingly small as the value becomes more extreme.

Probability Density Function (PDF):

The probability density function (PDF) of the exponential distribution is a function that describes the probability of a given value occurring under the distribution. The PDF of the exponential distribution is defined as:

$$f(x) = \lambda e^{-\lambda x}$$

where:

\(\lambda\) is the rate at which events occur

Cumulative Density Function (CDF)

The cumulative density function (CDF) of the exponential distribution is a function that describes the probability of a value being less than or equal to a given value. The CDF of the exponential distribution is defined as:

$$F(x) = 1 - e^{-\lambda x}$$

where:

\(\lambda\) is the rate at which events occur.

Mean, Median and Variance of Exponential Distribution:

Mean: The mean of the exponential distribution is equal to the inverse of the rate parameter \(\lambda\).

$$\mu = \frac{1}{\lambda}$$

Median: The median of the exponential distribution is equal to the inverse of the rate parameter $\lambda$ times the natural logarithm of 2.

$$\text{median} = \frac{\ln(2)}{\lambda}$$

Variance: The variance of the exponential distribution is equal to the inverse of the rate parameter squared.

$$\sigma^2 = \frac{1}{\lambda^2}$$

Standard deviation: The standard deviation of the exponential distribution is equal to the inverse of the rate parameter.

$$\sigma = \frac{1}{\lambda}$$

These are the mean, median, standard deviation, and variance of the exponential distribution. They are all related to the rate parameter $\lambda$ and can be used to describe the shape and spread of the distribution.

Using Microsoft Excel:

EXPON.DIST:

The EXPON.DIST function in Microsoft Excel calculates the probability density function (PDF) or cumulative distribution function (CDF) of the exponential distribution. The function has the following syntax:

EXPON.DIST(x, lambda, cumulative)

Where:

  • x: is the value for which you want to calculate the probability density or cumulative probability.
  • lambda: is the rate at which events occur in the exponential distribution.
  • cumulative: is a logical value that specifies whether you want to calculate the PDF (FALSE) or CDF (TRUE) of the exponential distribution.

The function returns the probability density or cumulative probability of the given value under the exponential distribution with the specified rate.

Example:

Suppose a machine has a failure rate of 0.5 failures per hour. What is the probability that the time between failures is less than 2 hours?

To solve this problem, we can use the EXPON.DIST function in Excel. The function takes three arguments: the value for which we want to calculate the probability (2 hours), the rate at which failures occur (0.5 failures/hour), and a logical value indicating whether we want the cumulative probability (TRUE).

In this case, the formula would be:

=EXPON.DIST(2, 0.5, TRUE)

This formula returns the cumulative probability that the time between failures is less than 2 hours, which is equal to 0.632121. This means that there is a 63.21% chance that the time between failures is less than 2 hours.

Conclusion:

The exponential distribution is a useful tool for modelling the time between events that occur at a constant rate. It has several important properties, including the memoryless property and a right-skewed shape.


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