Continuous Distributions Normal Beta Exponential Gamma Lognormal Weibull Uniform Triangular General

 Discrete Distributions Continuous Distributions Linear Transform

 Continuous Distributions - Normal

The probability density function of the normal distribution has the familiar "bell shape." It has two parameters and , the mean and standard deviation. The Normal distribution has applications in many practical contexts. It is often used, with theoretical justification, as the experimental variability associated with physical measurements. Many other distributions can be approximated by the normal distribution using suitable parameters. This distribution reaches its maximum value at , and it is symmetric about the mean, so the mode and the median also have the value . The figure plots the distribution associated with the example.

It is impossible to symbolically integrate the density function of a Normal distribution, so there is no closed form expression of the cumulative distribution function. The mean and standard deviation of the distribution are the parameters. The skewness is 0 and the kurtosis is 0.

Example

Example: Consider the problem of scheduling an operating room. From past experience we have observed that the expected time required for a single operation is 2.5 hours with a standard deviation of 1 hour. We decide to schedule the time for an operation so that there is a 90% chance that the operation will be finished in the allotted time. The question is how much time to allow? We assume the time required is a random variable with a normal distribution.

The figure shows the definition of the distribution created with the add-in. The moments are computed with user-defined functions. The entry in cell H9 is the solution to the problem. It holds the function:

RV_Inverse(RV14, 0.9)

The function returns the value of the random variable where the cumulative distribution is 0.9. Allowing 3.78 hours for an operation provides a 90% chance that the operation will be complete.

When numerical procedures are not available, probabilities are computed using tables that appear in many text books.

Certain well known probabilities associated with the normal distribution are in the table.

 Range Formula Probability Within one standard deviation of the mean 0.6826 Within two standard deviations of the mean 0.9545 Within three standard deviations of the mean 0.9973

Operations Research Models and Methods
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by Paul A. Jensen