> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-2-univariate-random-variables/11-gamma.md). # Gamma Gamma RV is a generalization of exponential RV that has domain $$\mathcal{D}=\[0,\infty)$$ and pdf: $$ f(x)=\frac{1}{\beta^\alpha \Gamma(\alpha)}e^{-x/\beta}x^{\alpha-1},; x\in \mathcal{D} $$ * Pdf has parameters $$\alpha,\beta\in \mathbb{R}\_+$$ * $$\Gamma(\alpha)$$ is the gamma function; for integer $$\alpha\geq 1$$ we have $$\Gamma(\alpha)=(\alpha-1)!$$. * We express fact that $$X$$ is a Gamma RV as $$X\sim \textrm{Gamma}(\alpha,\beta)$$ * For integer $$\alpha$$, the cdf of the Gamma RV takes the form: $$ F(x)=1-\sum\_{k=0}^{\alpha-1}\frac{1}{k!}\left(\frac{x}{\beta}\right)^ke^{-x/\beta} $$ * In context of time phenomena, RV generalizes exponential in that it models amount of time that we have to wait until we observe the $$\alpha$$-th occurrence of an event. * Consequently, $$\alpha$$=1 means first event (as in exponential RV). * This RV has applications not only in temporal but also in spatial phenomena. * For instance, it can be used to model distance traveled until we find $$\alpha$$-th occurrence of certain type of atom in a chain or crack in a pipeline. * Here are pdfs and cdfs for $$\textrm{Gamma}(\alpha,\beta)$$ for different values of $$\alpha,\beta$$.
* Note same as exponential for $$\alpha=1$$. * Note emergence of peaks due to competing effects for $$\alpha=2$$.
Python: reproduce this figure (ch2_compare_gammas.ipynb) Gamma RVs $$X\sim\mathrm{Gamma}(\alpha,\beta)$$ (Victor Z and J E Umaña, UW-Madison, 2024). `scipy.stats.gamma` uses shape `a` $$=\alpha$$ and `scale` $$=\beta$$. **Note:** This notebook reproduces the figure of the Gamma RV in Section 2.2.5 of *Statistics for Chemical Engineers*, with realizations of Gamma(1,1) and Gamma(2,1). ```python import numpy as np import matplotlib.pyplot as plt from scipy.stats import gamma alpha = [1, 2, 1, 2] beta = [1, 1, 0.5, 0.5] lw = [0.5, 1, 1.5, 2] x = np.linspace(0, 4, 1000) np.random.seed(0) S = 1000 plt.figure(figsize=(10, 8)) for sub, a_k in ((1, 1), (2, 2)): x_omega = gamma.rvs(a_k, scale=1, size=S) plt.subplot(2, 2, sub) plt.plot(x_omega, 'ko', markerfacecolor='w', markersize=4) plt.xlabel(r'Outcome $\omega$', fontsize=14) plt.ylabel(r'$x_\omega$', fontsize=14) plt.title(rf'$\alpha={a_k},\ \beta=1$') plt.grid(True) plt.axis([0, S, 0, 10]) for sub, fun, label in ((3, gamma.pdf, '$f(x)$'), (4, gamma.cdf, '$F(x)$')): plt.subplot(2, 2, sub) for k in range(len(alpha)): plt.plot(x, fun(x, a=alpha[k], scale=beta[k]), 'k', linewidth=lw[k], label=rf'$\alpha={alpha[k]},\ \beta={beta[k]}$') plt.grid(True) plt.xlabel('$x$', fontsize=14) plt.ylabel(label, fontsize=14) plt.legend(loc='lower right', fontsize=9) plt.tight_layout() plt.show() ```
**Note:** The Gamma RV generalizes the exponential RV (alpha = 1): it models the waiting time until the alpha-th event. Open in Colab: [`ch2_compare_gammas.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_compare_gammas.ipynb) · MATLAB live script: [`ch2_compare_gammas.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch2_compare_gammas.mlx)
## Example: Service Time {% hint style="danger" icon="flask" %} * You run warehouse logistics for your company and are responsible for making sure that queue of electric forklift trucks at charging stations remains under control. * Average charging rate at a station is 0.25 trucks/hr (average charge time is 4 hours). * Charge time is random because there is variability in state-of-charge and age of batteries and average charge time is quite long because you need to protect them. You are interested in addressing the following questions: {% endhint %} {% hint style="info" %} * If queue has 2 trucks, what is prob that next truck will wait more than 3 hours? * What is prob if queue has 3 trucks? * What is prob of waiting more than 4 hours if queue has 1 truck? * You buy brand new batteries and this allows you to cut down average charging time to 2 hours. What is prob of waiting more than 3 hours if queue has 1 truck? {% endhint %} {% hint style="danger" icon="flask" %} * Use $$X$$ to denote waiting time of next truck and let Poisson event represent completion of truck charging. * Prob that waiting time for next truck is more than 3 hours is $$\mathbb{P}(X>3)$$. * This is the prob that the charging of the 2 trucks in the queue *cannot* be completed in the next 3 hours. * $$X$$ is thus time to wait until we observe 2 Poisson events (two charge completions) {% endhint %} {% hint style="danger" icon="flask" %} * Average waiting time per event is $$\beta=1/0.25=4$$ hr. We use: $$ \mathbb{P}(X>x)=1-F(x)=\sum\_{k=0}^{\alpha-1}\frac{1}{k!}\left(\frac{x}{\beta}\right)^ke^{-x/\beta} $$ with $$x=3$$ (hr), $$\alpha=2$$, and $$\beta=4$$ (hr) and thus: $$ \begin{aligned} \mathbb{P}(X>3)&=\sum\_{k=0}^{1}\frac{1}{k!}\left(\frac{3}{4}\right)^ke^{-3/4}\\ &=\frac{1}{0!}\left(\frac{3}{4}\right)^0e^{-3/4}+\frac{1}{1!}\left(\frac{3}{4}\right)^1e^{-3/4}=e^{-3/4}+(3/4)e^{-3/4}\\ &=0.826. \end{aligned} $$ * If there are 3 trucks in queue we expect a higher probability. Indeed, set $$\alpha=3$$: $$ \begin{aligned} \mathbb{P}(X>3)&=\sum\_{k=0}^{2}\frac{1}{k!}\left(\frac{3}{4}\right)^ke^{-3/4}=0.959. \end{aligned} $$ {% endhint %} #### Python implementation {% hint style="success" %} Notebook [`ch2_example_service_time.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_example_service_time.ipynb) — open it in Colab and run the cells alongside this section. {% endhint %} Waiting time at a charging station as a Gamma RV (written for the book to accompany the slide example). With $$\alpha$$ trucks in the queue and average charge time $$\beta$$, $$ \mathbb{P}(X>x)=1-F(x)=\sum\_{k=0}^{\alpha-1}\frac{1}{k!}\left(\frac{x}{\beta}\right)^k e^{-x/\beta}. $$ Python names: `alpha`, `beta`, `x`; `P_exceed(x, alpha, beta)` implements the sum above. **Note:** This notebook reproduces Example 2.17 (Section 2.2.5) of *Statistics for Chemical Engineers*: the waiting time at a charging station modeled as a Gamma RV. ```python from math import exp, factorial from scipy.stats import gamma def P_exceed(x, alpha, beta): """P(X > x) for X ~ Gamma(alpha, beta) with integer alpha.""" return sum((x/beta)**k * exp(-x/beta) / factorial(k) for k in range(alpha)) cases = [ # (description, x [hr], alpha [trucks], beta [hr]) ("2 trucks in queue, wait > 3 hr", 3, 2, 4), ("3 trucks in queue, wait > 3 hr", 3, 3, 4), ("1 truck in queue, wait > 4 hr", 4, 1, 4), ("1 truck, new batteries (beta = 2 hr), wait > 3 hr", 3, 1, 2), ] for text, x, alpha, beta in cases: print(f"{text:52s} P = {P_exceed(x, alpha, beta):.3f} " f"(scipy: {gamma.sf(x, alpha, scale=beta):.3f})") ``` ``` 2 trucks in queue, wait > 3 hr P = 0.827 (scipy: 0.827) 3 trucks in queue, wait > 3 hr P = 0.959 (scipy: 0.959) 1 truck in queue, wait > 4 hr P = 0.368 (scipy: 0.368) 1 truck, new batteries (beta = 2 hr), wait > 3 hr P = 0.223 (scipy: 0.223) ``` **Note:** The more trucks in the queue, the more likely a long wait; faster charging (smaller beta) reduces it.