The cumulant generating function of gamma distribution is $K_X(t) =-\alpha \log \big(1-\beta t\big)$. ©2020 Matt Bognar Department of Statistics and Actuarial Science University of Iowa Hope you like Gamma Distribution article with step by step guide on various statistics properties of gamma probability. Let $X_1$ and $X_2$ be two independent Gamma variate with parameters $(\alpha_1, \beta)$ and $(\alpha_2, \beta)$ respectively. In this tutorial, we are going to discuss various important statistical properties of gamma distribution like graph of gamma distribution for various parameter combination, derivation of mean, variance, harmonic mean, mode, moment generating function and cumulant generating function. of gamma distribution with parameter $\alpha$ and $\beta$ is, $$ \begin{equation*} f(x) = \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta},\; x > 0;\alpha, \beta > 0 \end{equation*} $$, $$ \begin{equation*} \log f(x) = \log\bigg(\frac{1}{\beta^\alpha\Gamma(\alpha)}\bigg)+(\alpha-1)\log x -\frac{x}{\beta}. Then the harmonic mean of $G(\alpha,\beta)$ distribution is $H=\beta(\alpha-1)$. of $Y$ is, $$ \begin{eqnarray*} M_Y(t) &=& E(e^{tY}) \\ &=& E(e^{t(X_1+X_2)}) \\ &=& E(e^{tX_1} e^{tX_2}) \\ &=& E(e^{tX_1})\cdot E(e^{tX_2})\\ & &\qquad (\because X_1, X_2 \text{ are independent })\\ &=& M_{X_1}(t)\cdot M_{X_2}(t)\\ &=& \big(1-\beta t\big)^{-\alpha_1}\cdot \big(1-\beta t\big)^{-\alpha_2}\\ &=& \big(1-\beta t\big)^{-(\alpha_1+\alpha_2)}. $X$ is as follows: $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{\Gamma(\beta)}x^{\beta -1}e^{-x}, & \hbox{$x>0;\beta >0$;} \\ 0, & \hbox{Otherwise.} Here, we will provide an introduction to the gamma distribution. The sum of two independent Gamma variates is also Gamma variate. }\text{ in } K_X(t)\\ &=& \alpha \beta^r(r-1)!, r=1,2,\cdots \end{eqnarray*} $$, $$ \begin{eqnarray*} k_1 &=& \alpha\beta =\mu_1^\prime \\ k_2 &=& \alpha\beta^2=\mu_2\\ k_3 &=& 2\alpha\beta^3=\mu_3\\ k_4 &=& 6\alpha\beta^4=\mu_4-3\mu_2^2\\ \Rightarrow \mu_4 &=& 3\alpha(2+\alpha)\beta^4. The gamma distribution is another widely used distribution. \end{cases} \end{align*} $$. Gamma distribution functions with online calculator and graphing tool. This website uses cookies to ensure you get the best experience on our site and to provide a comment feature. A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.d.f. The variance of gamma distribution $G(\alpha,\beta)$ is $\alpha\beta^2$. $$ \begin{equation*} \frac{d^2\log f(x)}{dx^2}= -\frac{(\alpha-1)}{x^2}<0. Let $Y=X_1+X_2$. \end{array} \right. Gamma Distribution Formula, where p and x are a continuous random variable. This is left as an exercise for the reader. }+\cdots\bigg)\\ \end{eqnarray*} $$, Thus the $r^{th}$ cumulant of gamma distribution is, $$ \begin{eqnarray*} k_r & =& \text{coefficient of } \frac{t^r}{r! The parameters of the gamma distribution define the shape of the graph. \end{equation*} $$ $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{\alpha^\beta \Gamma(\beta)} x^{\beta -1}e^{-\frac{x}{\alpha}}, & \hbox{$x>0;\alpha, \beta >0$;} \\ 0, & \hbox{Otherwise.} of Gamma variate with parameter $(\alpha_1+\alpha_2, \beta)$. The $r^{th}$ raw moment of gamma distribution is, $$ \begin{eqnarray*} \mu_r^\prime &=& E(X^r) \\ &=& \int_0^\infty x^r\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\int_0^\infty x^{\alpha+r -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\Gamma(\alpha+r)\beta^{\alpha+r}\\ &=& \frac{\beta^r\Gamma(\alpha+r)}{\Gamma(\alpha)} \end{eqnarray*} $$. The moment generating function of gamma distribution is, $$ \begin{eqnarray*} M_X(t) &=& E(e^{tX}) \\ &=& \int_0^\infty e^{tx}\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\int_0^\infty x^{\alpha -1}e^{-(1/\beta-t) x}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\frac{\Gamma(\alpha)}{\big(\frac{1}{\beta}-t\big)^\alpha}\\ &=& \frac{1}{\beta^\alpha}\frac{\beta^\alpha}{\big(1-\beta t\big)^\alpha}\\ &=& \big(1-\beta t\big)^{-\alpha}, \text{ (if $t<\frac{1}{\beta}$}) \end{eqnarray*} $$. Gamma distribution is widely used in science and engineering to model a skewed distribution. where for $\alpha>0$, $\Gamma(\alpha)=\int_0^\infty x^{\alpha-1}e^{-x}; dx$ is called a gamma function. Calculates a table of the probability density function, or lower or upper cumulative distribution function of the gamma distribution, and draws the chart. © VrcAcademy - 2020About Us | Our Team | Privacy Policy | Terms of Use. Use given below Gamma Distribution Calculator to calculate probabilities with solved examples. is given by, $$ \begin{align*} f(x)&= \begin{cases} \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}, & x > 0;\alpha, \beta > 0; \\ 0, & Otherwise. In particular, we have the same basic shapes as given in Exercise 6. The gamma distribution with parameters k = 1 and b is called the exponential distribution with scale parameter b (or rate parameter r = 1 b). Therefore, mode of gamma distribution is $\beta(\alpha-1)$. The $r^{th}$ raw moment of gamma distribution is $\mu_r^\prime =\frac{\beta^r\Gamma(\alpha+r)}{\Gamma(\alpha)}$. The mean of gamma distribution $G(\alpha,\beta)$ is $\alpha\beta$. Gamma distribution is widely used in science and engineering to model a skewed distribution. Gamma distribution is used to model a continuous random variable which takes positive values. He holds a Ph.D. degree in Statistics. If $H$ is the harmonic mean of $G(\alpha,\beta)$ distribution then, $$ \begin{eqnarray*} \frac{1}{H}&=& E(1/X) \\ &=& \int_0^\infty \frac{1}{x}\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\int_0^\infty x^{\alpha-1 -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\Gamma(\alpha-1)\beta^{\alpha-1}\\ &=& \frac{1}{\beta^\alpha(\alpha-1)\Gamma(\alpha-1)}\Gamma(\alpha-1)\beta^{\alpha-1}\\ &=& \frac{1}{\beta(\alpha-1)}\\ & & \quad (\because\Gamma(\alpha) = (\alpha-1) \Gamma(\alpha-1)) \end{eqnarray*} $$, Therefore, harmonic mean of gamma distribution is, $$ \begin{equation*} H = \beta(\alpha-1). \end{eqnarray*} $$. Gamma Distribution. To understand more about how we use cookies, or for information on how to change your cookie settings, please see our Privacy Policy. Plus Four Confidence Interval for Proportion Examples, Weibull Distribution Examples - Step by Step Guide. The cumulant generating function of gamma distribution is, $$ \begin{eqnarray*} K_X(t)& = & \log_e M_X(t)\\ &=& \log_e \big(1-\beta t\big)^{-\alpha}\\ &=&-\alpha \log \big(1-\beta t\big)\\ &=& \alpha\big(\beta t +\frac{\beta^2 t^2}{2}+\frac{\beta^3 t^3}{3}+\cdots +\frac{\beta^r t^r}{r}+\cdots\big)\\ & & \qquad (\because \log (1-a) = -(a+\frac{a^2}{2}+\frac{a^3}{3}+\cdots))\\ &=& \alpha\bigg(t\beta+\frac{t^2\beta^2}{2}+\cdots +\frac{t^r\beta^r (r-1)!}{r! Hence, by Uniqueness theorem of m.g.f. Thus, variance of gamma distribution $G(\alpha,\beta)$ are $\mu_2 =\alpha\beta^2$. The mode of $G(\alpha,\beta)$ distribution is $\beta(\alpha-1)$. Gamma Distribution Graph. Notes about Gamma Distributions: If \(\alpha = 1\), then the corresponding gamma distribution is given by the exponential distribution, i.e., \(\text{gamma}(1,\lambda) = \text{exponential}(\lambda)\). Following is the graph of probability density function (pdf) of gamma distribution with parameter $\alpha=1$ and $\beta=1,2,4$. The moment generating function of gamma distribution is $M_X(t) =\big(1-\beta t\big)^{-\alpha}$ for $t< \frac{1}{\beta}$. Its importance is largely due to its relation to exponential and normal distributions. \end{equation*} $$, Differentiating $\log f(x)$ w.r.t. \end{eqnarray*} $$. ... graph horizontally and vertically. Another form of gamma distribution is Raju is nerd at heart with a background in Statistics. It is clear from the $\beta_1$ coefficient of skewness and $\beta_2$ coefficient of kurtosis, that, as $\alpha\to \infty$, $\beta_1\to 0$ and $\beta_2\to 3$. \end{eqnarray*} $$, The coefficient of skewness of gamma distribution is, $$ \begin{eqnarray*} \beta_1 &=& \frac{\mu_3^2}{\mu_2^3} \\ &=& \frac{(2\alpha\beta^3)^2}{(\alpha\beta^2)^3}\\ &=& \frac{4}{\alpha} \end{eqnarray*} $$, The coefficient of kurtosis of gamma distribution is, $$ \begin{eqnarray*} \beta_2 &=& \frac{\mu_4}{\mu_2^2} \\ &=& \frac{3\alpha(2+\alpha)\beta^4}{(\alpha\beta^2)^2}\\ &=& \frac{6+3\alpha}{\alpha} \end{eqnarray*} $$. \end{equation*} $$. \end{equation*} $$. $$ \begin{eqnarray*} \mu_2^\prime&= &E(X^2)\\ &=& \int_0^\infty x^2\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\int_0^\infty x^{\alpha+2 -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\Gamma(\alpha+2)\beta^{\alpha+2}\\ & & \quad (\text{using gamma integral})\\ &=& \alpha(\alpha+1)\beta^2,\\ & & \quad (\because\Gamma(\alpha+2) = (\alpha+1) \alpha\Gamma(\alpha)) \end{eqnarray*} $$, Hence, the variance of gamma distribution is, $$ \begin{eqnarray*} \text{Variance = } \mu_2&=&\mu_2^\prime-(\mu_1^\prime)^2\\ &=&\alpha(\alpha+1)\beta^2 - (\alpha\beta)^2\\ &=&\alpha\beta^2(\alpha+1-\alpha)\\ &=&\alpha\beta^2. We can use the Gamma distribution for every application where the exponential distribution is used — Wait time modeling, Reliability (failure) modeling, Service time modeling (Queuing Theory), etc.