Sum of exponential random variables follows Gamma, confused by the parameters - Cross Validated most recent 30 from stats.stackexchange.com 2019-07-17T11:25:26Z https://stats.stackexchange.com/feeds/question/27908 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/27908 17 Sum of exponential random variables follows Gamma, confused by the parameters edwin https://stats.stackexchange.com/users/11138 2012-05-06T20:42:15Z 2018-04-22T05:46:36Z <p>I've learned sum of exponential random variables follows Gamma distribution.</p> <p>But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?</p> <p>Exponential distribution: $x$~$exp(\lambda)$ $$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$ $$E[x]=1/ \lambda$$ $$var(x)=1/{{\lambda}^2}$$</p> <p>Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$ $$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$ $$E[x]=\alpha\beta$$ $$var[x]=\alpha{\beta}^{2}$$</p> <p>In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?</p> https://stats.stackexchange.com/questions/27908/-/27909#27909 10 Answer by Neil G for Sum of exponential random variables follows Gamma, confused by the parameters Neil G https://stats.stackexchange.com/users/858 2012-05-06T21:57:42Z 2012-05-06T21:57:42Z <p>The sum of $n$ iid exponential distributions with scale $\theta$ (rate $\theta^{-1}$) is gamma-distributed with shape $n$ and scale $\theta$ (rate $\theta^{-1}$).</p> https://stats.stackexchange.com/questions/27908/-/27910#27910 12 Answer by Dilip Sarwate for Sum of exponential random variables follows Gamma, confused by the parameters Dilip Sarwate https://stats.stackexchange.com/users/6633 2012-05-06T22:02:01Z 2015-09-25T19:47:12Z <p>The sum of $n$ independent Gamma random variables $\sim \Gamma(t_i, \lambda)$ is a Gamma random variable $\sim \Gamma\left(\sum_i t_i, \lambda\right)$. It does not matter what the second parameter means (scale or inverse of scale) as long as all $n$ random variable have the <em>same</em> second parameter. This idea extends readily to $\chi^2$ random variables which are a special case of Gamma random variables.</p> https://stats.stackexchange.com/questions/27908/-/164449#164449 1 Answer by hasanmisaii for Sum of exponential random variables follows Gamma, confused by the parameters hasanmisaii https://stats.stackexchange.com/users/83971 2015-08-03T08:22:27Z 2015-09-25T20:30:38Z <p>gamma distribution is made of exponential distribution that is exponential distribution is base for gamma distribution. then if $f(x|\lambda)=\lambda e^{−\lambda x}$ we have $\sum_n x_i \sim \text{Gamma}(n,\lambda)$, as long as all $X_i$ are independent.</p> <p>$$f(x|\alpha,\beta)=\frac{\beta^α}{\Gamma(\alpha)} \cdot x^{\alpha−1} \cdot e^{−x\beta}$$ </p>