Gamma distribution explain

Question

what is Gamma distribution, can anyone explain to me? I am confused when reciting the Gamma density formula $\dfrac{\exp(-\frac{x}{\sigma})x^{a-1}}{\sigma^a \Gamma \left( a \right)}, 0<x<\infty$

Answer 1

My own confusions about recalling the density were cleared up when I wrote it like this: \begin{align} & \frac 1 {\Gamma(a)}\left( \frac x \sigma \right)^{a-1} e^{-x/\sigma} \left( \frac{dx} \sigma \right) & & \text{for } x\ge 0 \\[10pt] = {} & \frac 1 {\Gamma(a)} u^{a-1} e^{-u} \, du & & \text{for } u \ge 0. \end{align} Thus the substitution $u = \dfrac x \sigma$ reduces this to the thing you integrate when you define the Gamma function. One can also write $$ \frac 1 {\Gamma(a)} u^a e^{-u} \,\, \frac{du} u \qquad \text{for } u\ge0. $$

Picture something like the "bell-shaped curve" that is the normal density, except that it is constrained to lie above $0$, so the left tail dies off completely when you reach $0$ and the right tail never fully vanishes as you go upward.

Here are some characterizations:

• When $a$ is a positive integer, then the gamma distribution is the distribution of the sum of $a$ independent random variables each of which has a memoryless exponential distribution with expected value $\sigma.$
• When $a$ is half of a positive integer, then the gamma distribution is the distribution of the sum of the squares of $2a$ independent random variables each having a normal distribution with expected value $0$ and variance $\sigma.$
• If $X,Y$ are independent random variables that are non-degenerately distributed and have positive values then $X+Y$ and $X/Y$ are independent only if $X,Y$ both have gamma distributions with the same scale parameter $\sigma.$ [Eugene Lukacs, The Annals of Mathematical Statistics vol. 26, No. 2 (June, 1955), pp. 319–324]
• For i.i.d. positive random variables $X_1,X_2,X_3,\ldots,$ the same mean $\overline X_n = (X_1+\cdots+X_n)/n$ and the sample coefficient of variation $S_n/\overline X_n$ are independent only if the common distribution is a gamma distribution. (I'm not sure if one needs to say "for all sample sizes $n$".) ["On a Characterization of the Gamma Distribution: The Independence of the Sample Mean and the Sample Coefficient of Variation", Tea-Yuan Hwang and Chin-Yuan Hu, Annals of the Institute of Statistical Mathematics, December 1999, Volume 51, Issue 4, pp. 749–753.]