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of gamma distribution with parameter $\alpha$ and $\beta$ is, $$ \begin{equation*} f(x) = \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta},\; x > 0;\alpha, \beta > 0 \end{equation*} $$, $$ \begin{equation*} \log f(x) = \log\bigg(\frac{1}{\beta^\alpha\Gamma(\alpha)}\bigg)+(\alpha-1)\log x -\frac{x}{\beta}. To find variance of $X$, we need to find $E(X^2)$. Gamma distribution is widely used in science and engineering to model a skewed distribution. In this tutorial, we are going to discuss various important statistical properties of gamma distribution like graph of gamma distribution for various parameter combination, derivation of mean, variance, harmonic mean, mode, moment generating function and cumulant generating function. Hope you like Gamma Distribution article with step by step guide on various statistics properties of gamma probability. To understand more about how we use cookies, or for information on how to change your cookie settings, please see our Privacy Policy. \end{equation*} $$. ... graph horizontally and vertically. He holds a Ph.D. degree in Statistics. Calculates a table of the probability density function, or lower or upper cumulative distribution function of the gamma distribution, and draws the chart. \end{equation*} $$. $x$ and equating to zero, we get, $$ \begin{eqnarray*} & & \frac{d\log f(x)}{dx}=0 \\ &\Rightarrow& 0+ \frac{\alpha-1}{x}-\frac{1}{\beta} =0\\ &\Rightarrow& x=\beta(\alpha-1). Thus, the $r^{th}$ raw moment of gamma distribution is $\mu_r^\prime =\frac{\beta^r\Gamma(\alpha+r)}{\Gamma(\alpha)}$. Gamma Distribution. $$ \begin{eqnarray*} \mu_2^\prime&= &E(X^2)\\ &=& \int_0^\infty x^2\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\int_0^\infty x^{\alpha+2 -1}e^{-x/\beta}\; dx\\ &=& \frac{1}{\beta^\alpha\Gamma(\alpha)}\Gamma(\alpha+2)\beta^{\alpha+2}\\ & & \quad (\text{using gamma integral})\\ &=& \alpha(\alpha+1)\beta^2,\\ & & \quad (\because\Gamma(\alpha+2) = (\alpha+1) \alpha\Gamma(\alpha)) \end{eqnarray*} $$, Hence, the variance of gamma distribution is, $$ \begin{eqnarray*} \text{Variance = } \mu_2&=&\mu_2^\prime-(\mu_1^\prime)^2\\ &=&\alpha(\alpha+1)\beta^2 - (\alpha\beta)^2\\ &=&\alpha\beta^2(\alpha+1-\alpha)\\ &=&\alpha\beta^2. Following is the graph of probability density function (pdf) of gamma distribution with parameter $\alpha=1$ and $\beta=1,2,4$. Hence, by Uniqueness theorem of m.g.f. For $\alpha =1$, gamma distribution $G(\alpha, \beta)$ becomes an exponential distribution with parameter $\beta$. The variance of gamma distribution $G(\alpha,\beta)$ is $\alpha\beta^2$. The mean of the gamma distribution $G(\alpha,\beta)$ is, The mean of $G(\alpha,\beta)$ distribution is, $$ \begin{eqnarray*} \text{mean = }\mu_1^\prime &=& E(X) \\ &=& \int_0^\infty 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}\\ & & \quad (\text{Using }\int_0^\infty x^{n-1}e^{-x/\theta}\; dx = \Gamma(n)\theta^n )\\ &=& \alpha\beta,\;\quad (\because\Gamma(\alpha+1) = \alpha \Gamma(\alpha)) \end{eqnarray*} $$. We can use the Gamma distribution for every application where the exponential distribution is used — Wait time modeling, Reliability (failure) modeling, Service time modeling (Queuing Theory), etc. Gamma distribution functions with online calculator and graphing tool. $Y=X_1+X_2$ is a Gamma variate with parameter $(\alpha_1+\alpha_2, \beta)$. In Chapters 6 and 11, we will discuss more properties of the gamma random variables. Hence, the density $f(x)$ becomes maximum at $x =\beta(\alpha-1)$. Here, we will provide an introduction to the gamma distribution. ©2020 Matt Bognar Department of Statistics and Actuarial Science University of Iowa is given by, $$ \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. Gamma Distribution Formula, where p and x are a continuous random variable. which is the m.g.f. which is one parameter gamma distribution. This website uses cookies to ensure you get the best experience on our site and to provide a comment feature. \end{equation*} $$, Differentiating $\log f(x)$ w.r.t. of Gamma variate with parameter $(\alpha_1+\alpha_2, \beta)$. The gamma distribution with parameters k = 1 and b is called the exponential distribution with scale parameter b (or rate parameter r = 1 b). Let $H$ be the harmonic mean of gamma distribution. \end{array} \right. The moment generating function of gamma distribution is, $$ \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*} $$. }+\cdots\bigg)\\ \end{eqnarray*} $$, Thus the $r^{th}$ cumulant of gamma distribution is, $$ \begin{eqnarray*} k_r & =& \text{coefficient of } \frac{t^r}{r! The $r^{th}$ raw moment of gamma distribution is $\mu_r^\prime =\frac{\beta^r\Gamma(\alpha+r)}{\Gamma(\alpha)}$. The cumulant generating function of gamma distribution is, $$ \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! Plus Four Confidence Interval for Proportion Examples, Weibull Distribution Examples - Step by Step Guide. Gamma distribution is used to model a continuous random variable which takes positive values. \end{eqnarray*} $$. Let $X_1$ and $X_2$ be two independent Gamma variate with parameters $(\alpha_1, \beta)$ and $(\alpha_2, \beta)$ respectively. Raju is nerd at heart with a background in Statistics. The parameter $\alpha$ is called the shape parameter and $\beta$ is called the scale parameter of gamma distribution. — because exponential distribution is a special case of Gamma distribution (just plug 1 into k). The cumulant generating function of gamma distribution is $K_X(t) =-\alpha \log \big(1-\beta t\big)$. © VrcAcademy - 2020About Us | Our Team | Privacy Policy | Terms of Use. 16. $$ \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.} \end{cases} \end{align*} $$. If $H$ is the harmonic mean of $G(\alpha,\beta)$ distribution then, $$ \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).

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