{"id":16406,"date":"2020-06-25T15:15:13","date_gmt":"2020-06-25T09:45:13","guid":{"rendered":"https:\/\/www.mygreatlearning.com\/blog\/gamma-distribution\/"},"modified":"2024-01-30T19:13:23","modified_gmt":"2024-01-30T13:43:23","slug":"gamma-distribution","status":"publish","type":"post","link":"https:\/\/www.mygreatlearning.com\/blog\/gamma-distribution\/","title":{"rendered":"What is Gamma Distribution?"},"content":{"rendered":"\n

Contributed by: Somak Sengupta <\/a><\/strong><\/p>\n\n\n\n

What is Gamma Distribution?<\/strong><\/h2>\n\n\n\n

Gamma Distribution is a Continuous Probability Distribution<\/strong> that is widely used in different fields of science to model continuous variables that are always positive and have skewed distributions<\/strong>. It occurs naturally in the processes where the waiting times between events are relevant. <\/p>\n\n\n\n

Why do we need Gamma Distribution?<\/strong><\/h2>\n\n\n\n

It is used to predict the wait time until future events occur. As we shall see the parameterization below, Gamma Distribution predicts the wait time until the k-th (Shape parameter) event occurs. <\/p>\n\n\n\n

The PDF of the Gamma Distribution<\/strong><\/h2>\n\n\n\n

It is a two-parameter continuous probability distribution. Exponential distribution and Chi-squared distribution are two of the special cases which we\u2019ll see how we can derive from the Gamma Distribution. The commonly used parameterization are as follows-<\/p>\n\n\n\n

It is a two-parameter continuous probability distribution. Exponential distribution and Chi-squared distribution are two of the special cases, the derivation of which from Gamma Distribution we will see. The commonly used parameterizations are as follows-<\/p>\n\n\n\n

    \n
  1. Shape parameter = k and Scale parameter = \u03b8.<\/li>\n\n\n\n
  2. Shape parameter \u03b1 = k and an Inverse Scale parameter \u03b2 = 1\/\u03b8 called a Rate parameter<\/strong>. In exponential distribution, we call it as \u03bb (lambda, \u03bb = 1\/\u03b8) which is known as the Rate of the Events<\/strong> happening that follows the Poisson process. While k is the number of events<\/strong> until which we are waiting for the expected event to occur.<\/li>\n\n\n\n
  3. Shape parameter = k and a Mean parameter \u03bc = k*\u03b8 = \u03b1\/\u03b2.<\/li>\n<\/ol>\n\n\n\n

    The general formulation for the probability density function<\/a> (PDF) is-<\/p>\n\n\n\n

    where, the Gamma Function<\/strong> is defined as - \u0393(\u03b1) = (\u03b1-1)!<\/strong> for all positive integers<\/strong>. The gamma function*<\/strong> is eventually derived from the following integral\u2013
    <\/p>\n\n\n\n

    *Note that Gamma Distribution and Gamma Function are two different concepts. <\/p>\n\n\n\n

    Using the parameters as k (# of events and k>0) and \u03b8 (\u03bb = 1\/\u03b8) where \u03bb is the rate of the event, we can write the PDF (Eq. 1) of the Gamma Distribution as\u2013<\/p>\n\n\n\n

    where \u0393(k) = (k-1)! <\/strong><\/p>\n\n\n\n

    Therefore, a random variable X is eventually denoted by- <\/p>\n\n\n\n

          ---Eq. 4<\/p>\n\n\n\n

    Special Case of Gamma Distribution \u2013 Exponential Distribution (k=1) <\/strong><\/p>\n\n\n\n

    Special Case of Gamma Distribution \u2013 Chi Squared Distribution (\u03b8=2, k=n\/2, n = degrees of freedom)<\/strong><\/p>\n\n\n\n

    Other Important things to know\u2013 Properties<\/strong><\/h4>\n\n\n\n

    Mean \u2013 k\u03b8
    Variance \u2013 k\u03b82<\/sup>
    Skewness \u2013 2\/sqrt(k)<\/p>\n\n\n\n

    Few important points<\/strong><\/h4>\n\n\n\n