Expectation

Sampling

Probability Density Function

Cumulative Distribution Function

Reliability Function

Hazard Function

Exam

Engineered Software Home Page

Several terms are used to describe distributions, among the most common are the mean, variance, skewness and kurtosis.  These descriptors are derived from moment generating functions.  Readers with an engineering background may recall that the center of gravity of a shape is

The mean, or average, of a distribution is its center of gravity.  In the equation above, the denominator is equal to the area below f(x), which is equal to one by definition for valid probability distributions.  The numerator in the equation above is the first moment generating function about the origin. Thus, the mean of a distribution can be determined from the expression

 The second moment generating function about the origin is

 The variance of a distribution is equal to the second moment generating function about the mean, which is

 The variance is a measure of the dispersion in a distribution.  In the figure below, the variance of distribution A is greater than the variance of distribution B.

Another common measure of dispersion is the standard deviation.  The standard deviation is equal to the square root of the variance.  The distribution standard deviation is estimated from the sample standard deviation

Note: The sample standard deviation can be found using the Excel function "=STDEV()".

The skewness of a distribution is equal to the third moment generating function about the mean. If the skewness is positive, the distribution is right skewed. If the skewness is negative, the distribution is left skewed. Right and left skewness are demonstrated in the figure below.

Kurtosis is the fourth moment generating function about the mean, and is a measure of the peakedness of the distribution.