Probability Distributions

    Distributions:
  1. Binomial Distribution
  2. Poisson Distribution
  3. Negative Binomial Distribution
  4. Geometric Distribution
  5. T Distribution
  6. Chi-squared Distribution
  7. Gamma Distribution
  8. Weibull Distribution
  9. Log-Normal Distribution
  10. Beta Distribution
  11. F Distribution
    Specialty Functions
  1. Gamma Function
  2. Beta Function

  1. The Binomial Distribution is used in finite sampling problems where each observation is one of two possible outcomes ("success" or "failure"). The binomial distribution has two parameters:

    1. n = the sample size, and
    2. = P("success").

    Example: To assure quality of a product, a random sample of size 25 is drawn from a process. The number of defects (X) found in the sample is recorded. The random variable X follows a binomial distribution with n = 25 and = P(product is defective).

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  2. The Poisson Distribution is used for modeling rates of occurrence. The Poisson distribution has one parameter:

    1. = the rate (mean).

    Example: A process that creates fabric is monitored. If the number of defects (X) per meter of fabric exceeds 5 then the process is stopped for diagnosis. The random variable X follows a Poisson distribution with = number of defects per meter of fabric.

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  3. The Negative Binomial Distribution is used for modeling rates of occurrence. The Negative Binomial distribution has two parameters:

    1. r = the total number of failures.
    2. = P("success").

    Example: A process that manufactures widgets is monitored. As each widget exits the process line, it is tested for defective versus non-defective. On the fifth defect, the process is stopped for re-adjustment. The random variable X follows a Negative Binomial distribution with r = 5 and = P(widget is non-defective).

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  4. The Geometric Distribution is used for modeling rates of occurrence. The Geometric distribution has one parameter:

    1. = P("success").

    Example: A process that manufactures widgets is monitored. As each widget exits the process line, it is tested for defective versus non-defective. On the first defect, the process is stopped for re-adjustment. The random variable X follows a Geometric distribution with = P(widget is non-defective).
    See also:     Negative Binomial.

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  5. The T Distribution is used in many situations, some of which are listed below. The T distribution has one parameter:
    1. = degrees of freedom.

    Common usage:
    • Inference on a single normal mean, variance unknown.
    • Inference on the comparison of two normal means, variance unknown.
    • Inference on individual regression parameters.

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  6. The Chi-squared Distribution is used in many situations, some of which are listed below. The Chi-squared distribution has one parameter:
    1. = degrees of freedom.

    Common usage:
    • Inference on a single normal variance.
    • Chi-squared Tests:
       - test for independence,
       - homogenity,
       - goodness of fit.

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  7. The Gamma Distribution is a general distribution covering many special cases, including the Chi-squared distribution and Exponential distribution. The Gamma distribution has two parameters:
    1. = rate parameter.
    2. = scale parameter.

    Common usage:
    • Positively skewed data such as movement data and electrical measurements.

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  8. The Weibull Distribution is typically used in reliability modeling. The Weibull distribution has two parameters:
    1. = rate parameter.
    2. = scale parameter.
    For = = 1, the Weibull distribution is identical to the Exponential distribution.

    Common usage:
    • Time to failure modeling.

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  9. The Log-Normal Distribution is useful when the raw data are highly skewed whereas the natural log of the data are normally distributed. The Log-Normal distribution has two parameters:
    1. = location parameter.
    2. = scale parameter.

    Common usage:
    • Positively skewed data such as movement data and electrical measurements.

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  10. The Beta Distribution is a continuous distribution bounded between 0 and 1. The Beta distribution has two parameters:
    1. a = first shape parameter.
    2. b = second shape parameter.
    When a = b = 1, the Beta distribution is identical to the Uniform distribution on (0,1).

    Common usage:
    • Modeling the probability of success for a binomial distribution.

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  11. The F Distribution is used in many situations, some of which are listed below. The F distribution has two parameters:
    1. 1 = numerator degrees of freedom,
    2. 2 = denominator degrees of freedom.

    Common usage:
    • Inference on two or more normal variances.
    • ANOVA.
    • Regression.

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Specialty Functions

  1. The Gamma function, used in the several of the distributions described above, is defined by
    (1)
    A special case of (1) is
    (2)

    where n is some integer. (2) is commonly used for integer degrees of freedom in the above distributions. However, it should be noted that even though (2) is the most common case (e.g. df = n - 1),there is no requirement for degrees of freedom to be integers.

  2. The Beta function, used only in the Beta distribution, is defined by

See also: Normal Distribution, T Distribution, Exponential Distribution.
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