INTERACTIVE ANIMATION: THE GAMMA DISTRIBUTION

This animation illustrates the Gamma distribution.

Foreword

In fact, we only illustrate the following special case of the Gamma distribution :

    * Let X_i (i = 1, 2, ...n) be n independent exponentially distributed random variables.

X_i ~ Exp()

Denote X the r.v. defined as the sum of the X_i :

X = _i X_i 

Then X is Gamma distributed with parameters n and :

X ~ Gamma(n, )

For a  proof, see the special case of the Gamma distribution when α is an integer.

This is a special case because α is here constrained to be a positive integer n, while in the general case, it can be any positive real number. The expression for the probability density function is now :

owing to the properties of the Gamma function.

_________________________________________

Upper frame

The n iid exponentially distributed r.v. are simulated by drawing a n-point sample from the single exponential distribution in the upper frame.

    Lambda

        The value of is just the height of the point where the exponential curve meets the vertical axis (intercept).

    Mean

        Recall that the mean of the distribution is 1/. It is displayed as a vertical blue line. Change the value of  by sliding the mean with your mouse positioned just over the top of the displayed value of the mean.

    Sample size

        Adjust n, the sample size, with the "Nb. Points" control buttons.

Lower frame

    Gamma distribution

        The red curve is the Gamma distribution for the selected values of  and n.

    Mean

        The mean value is n/, and is displayed as a vertical blue line.

    Mode

        We leave as an exercise to show that the mode is (n-1)/. It is therefore always less than the mean, as expected from a positively skewed bell-shaped distribution.

    Special case n = 1

        When n = 1, the Gamma distribution is just the exponential distribution Exp().

Animation

    After choosing the values of  and n, click on "Go" and watch the build-up of the histogram of the corresponding Gamma distribution.

Note that in the "Next" mode, the value of X is displayed between the upper and lower frames.
 

Warning : the horizontal and vertical scales are different in the upper and lower frames.