## F.2 Continuous Distrbutions

Uniform $$U(a,b)$$
Parameters: a = minimum, b = maximum, $$-\infty < a < b < \infty$$
PDF: $$f(x) = \frac{1}{b-a}$$ for $$a \leq x \leq b$$
CDF: $$F(x) = \frac{x-a}{b-a} \; \text{if} \; a \leq x \leq b$$
Inverse CDF: $$F^{-1}(p) = a + p(b-a) \; \; \text{if} \; 0 < p < 1$$
Expected Value: $$E[X]=\frac{a+b}{2}$$
Variance: $$V[X] = \frac{(b-a)^2}{12}$$
Arena Generation: UNIF(a,b[,Stream])
Spreadsheet Generation: = a + RAND()*(b-a)
Modeling: assumes equally likely across the range,
when you have lack of data, task times
Normal $$N(\mu,\sigma^2)$$
Parameters: $$-\infty < \mu < +\infty$$ (mean), $$\sigma^2 > 0$$ (variance)
CDF: No closed form
Inverse CDF: No closed form
Expected Value: $$E[X] = \mu$$
Variance: $$Var[X] = \sigma^2$$
Arena Generation: NORM($$\mu,\sigma^2$$[,Stream])
Spreadsheet Generation: = NORM.INV(RAND(), $$\mu$$, $$\sigma$$)
Exponential EXPO($$1/\lambda$$)
Parameters: $$\lambda > 0$$
PDF: $$f(x) = \lambda e^{-\lambda x} \; \text{if} \; x \geq 0$$
CDF: $$F(x) = 1 - e^{-\lambda x} \; \text{if} \; x \geq 0$$
Inverse CDF: $$F^{-1}(p) = (-1/\lambda)\ln \left(1-p \right) \; \; \text{if} \; 0 < p < 1$$
Expected Value: $$E[X] = \theta = 1/\lambda$$
Variance: $$Var[X] = 1/\lambda^2$$
Arena Generation: EXPO($$\theta$$[,Stream])
Spreadsheet Generation: = $$(-1/\lambda)$$LN(1-RAND())
Modeling: time between arrivals, time to failure
Weibull WEIB($$\beta$$, $$\alpha$$)
Parameters: $$\beta > 0$$ (scale), $$\alpha > 0$$ (shape)
CDF: $$F(x) = 1- e^{-(x/\beta)^\alpha} \; \text{if} \; x \geq 0$$
Inverse CDF: $$F^{-1}(p) = \beta\left[ -\ln (1-p)\right]^{1/\alpha} \; \; \text{if} \; 0 < p < 1$$
Expected Value: $$E[X] = \left(\dfrac{\beta}{\alpha}\right)\Gamma\left(\dfrac{1}{\alpha}\right)$$
Variance: $$Var[X] = \left(\dfrac{\beta^2}{\alpha}\right)\biggl\lbrace 2\Gamma\left(\dfrac{2}{\alpha}\right) - \left(\dfrac{1}{\alpha}\right)\biggl(\Gamma\left(\dfrac{1}{\alpha}\right)\biggr)^2\biggr\rbrace$$
Arena Generation: WEIB(scale, shape[,Stream])
Spreadsheet Generation: = $$(\beta)(-\text{LN}(1-\text{RAND}())\wedge(1/\alpha)$$
Modeling: task times, time to failure
Erlang Erlang($$r$$,$$\beta$$)
Parameters: $$r > 0$$, integer, $$\beta > 0$$ (scale)
CDF: $$F(x) = 1- e^{(-x/\beta)}\sum\limits_{j=0}^{r-1}\dfrac{(x/\beta)^j}{j} \; \text{if} \; x \geq 0$$
Inverse CDF: No closed form
Expected Value: $$E[X] = r\beta$$
Variance: $$Var[X] = r\beta^2$$
Arena Generation: ERLA($$E[X], r$$[,Stream])
Spreadsheet Generation: = GAMMA.INV(RAND(), $$r$$, $$\beta$$)
Gamma Gamma($$\alpha$$,$$\beta$$)
Parameters: $$\alpha > 0$$, shape, $$\beta > 0$$ (scale)
CDF: No closed form
Inverse CDF: No closed form
Expected Value: $$E[X] = \alpha \beta$$
Variance: $$Var[X] = \alpha \beta^2$$
Arena Generation: GAMM(scale, shape[,Stream])
Spreadsheet Generation: = GAMMA.INV(RAND(), $$\alpha$$, $$\beta$$)
Beta BETA($$\alpha_1$$,$$\alpha_2$$)
Parameters: shape parameters $$\alpha_1 >0$$, $$\alpha_2 >0$$
CDF: No closed form
Inverse CDF: No closed form
Expected Value: $$E[X] = \dfrac{\alpha_1}{\alpha_1 + \alpha_2}$$
Variance: $$Var[X] = \dfrac{\alpha_1\alpha_2}{(\alpha_1 + \alpha_2)^2(\alpha_1 + \alpha_2+1)}$$
Arena Generation: BETA($$\alpha_1$$,$$\alpha_2$$[,Stream])
Spreadsheet Generation: BETA.INV(RAND(), $$\alpha_1$$, $$\alpha_2$$)
Modeling: activity time when data is limited, probabilities
Lognormal LOGN$$\left(\mu_l,\sigma_l\right)$$
Parameters: $$\mu = \ln\left(\mu_{l}^{2}/\sqrt{\sigma_{l}^{2} + \mu_{l}^{2}}\right) \quad \sigma^{2} = \ln\left((\sigma_{l}^{2}/\mu_{l}^{2}) + 1\right)$$
CDF: No closed form
Inverse CDF: No closed form
Expected Value: $$E[X] = \mu_l = e^{\mu + \sigma^{2}/2}$$
Variance: $$Var[X] = \sigma_{l}^{2} = e^{2\mu + \sigma^{2}}\left(e^{\sigma^{2}} - 1\right)$$
Arena Generation: LOGN($$\mu_l,\sigma_l$$[,Stream])
Spreadsheet Generation: LOGNORM.INV(RAND(), $$\mu$$, $$\sigma$$)
Modeling: task times, time to failure
Triangular TRIA(a, m, b)
Parameters: a = minimum, m = mode, b = maximum
CDF: $$F(x) = \dfrac{(x - a)^2}{(b - a)(m - a)} \; \text{for} \; a \leq x \leq m$$
$$F(x) = 1 - \dfrac{(b - x)^2}{(b - a)(b - m)} \; \text{for} \;m < x \leq b$$
Inverse CDF: $$F^{-1}(u) = a + \sqrt{(b-a)(m-a)u} \; \text{for} \; 0 < u < \dfrac{m-a}{b-a}$$
$$F^{-1}(u) = b - \sqrt{(b-a)(b-m)(1-u)} \; \text{for} \; \dfrac{m-a}{b-a} \leq u$$
Expected Value: $$E[X] = (a+m+b)/3$$
Variance: $$Var[X] = \dfrac{a^2 + b^2 + m^2 -ab -am -bm}{18}$$
Arena Generation: TRIA(min, mode, max[,Stream])
Spreadsheet Generation: implement $$F^{-1}(u)$$ as VBA function
Modeling: task times, activity time when data is limited