Clear thinking and simplicity of analysis require concise, clear, and correct notions about probability models and how to use them. Here, we’ll examine the basic properties of the family of gamma and chi-square distributions that play major roles in the development of statistical techniques. An appreciation of how these probability models function will allow you to analyze your data with confidence that your results are reliable, solid, and correct.

The gamma family of distributions

Gamma distributions are widely used in all areas of statistics, and are found in most statistical software. Since software facilitates our use of the gamma models, the following formulas are given here only in the interest of clarity of notation. Gamma distributions depend upon two parameters, denoted here by alpha and beta. The probability density function for the gamma family has the form:

where the symbol Γ(α) denotes the gamma function (for α > 0):

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