## F.1 Discrete Distrbutions

Bernoulli $$Ber(p)$$
Parameters: $$0 < p < 1$$, probability of success
PMF: $$P[X=1] = p, \; P[X=0] = 1-p$$
Inverse CDF: $$F^{-1}(u) = \text{if}\ (u < p), 1 \ \text{else} \ 0$$
Expected Value: $$E[X] = p$$
Variance: $$Var[X] = p(1-p)$$
Arena Generation: DISC(p,1,1.0,0[,Stream])
Spreadsheet Generation: = IF(RAND()$$<$$ p, 1, 0)
Modeling: the number of successes in one trial
Binomial $$Binom(n,p)$$
Parameters: $$0 < p < 1$$, probability of success, $$n$$, number of trials
PMF: $$P[X=x] = \binom{n}{x}p^{x}(1-p)^{n-x} \quad x=0,1,\ldots,n$$
Inverse CDF: no closed form available
Expected Value: $$E[X] = np$$
Variance: $$Var[X] = np(1-p)$$
Arena Generation: not available, use convolution of Bernoulli
Modeling: the number of successes in $$n$$ trials
Shifted Geometric Shifted Geo($$p$$)
Parameters: $$0 < p < 1$$, probability of success
PMF: $$P[X=x] = p(1-p)^{x-1} \quad x=1,2,\ldots,$$
Inverse CDF $$F^{-1}(u) = 1 + \left\lfloor \frac{ln(1 - u)}{ln(1 - p)} \right\rfloor$$
Expected Value: $$E[X] = 1/p$$
Variance: $$Var[X] = (1-p)/p^2$$
Arena Generation: 1 + AINT(LN(1-UNIF(0,1))/LN(1-p))
Spreadsheet Generation: = $$\text{1 + INT(LN(1-RAND())/LN(1-p))}$$
Modeling: the number of trials until the first success
Negative Binomial Defn. 1 NB1($$r,p$$)
Parameters: $$0 < p < 1$$, probability of success, $$r^{th}$$ success
PMF: $$P[X=x] = \binom{x-1}{r-1}p^{r}(1-p)^{x-r} \quad x=r,r+1\ldots,$$
Inverse CDF: no closed form available
Expected Value: $$E[X] = r/p$$
Variance: $$Var[X] = r(1-p)/p^2$$
Arena Generation: use convolution of shifted geometric
Spreadsheet Generation: use convolution of shifted geometric
Modeling: the number of trials until the $$r^{th}$$ success
Negative Binomial Defn. 2 NB2($$r,p$$)
Parameters: $$0 < p < 1$$, probability of success, $$r^{th}$$ success
PMF: $$P[Y=y] = \binom{y+r-1}{r-1}p^{r}(1-p)^{y} \quad y=0,1,\ldots$$
Inverse CDF: no closed form available
Expected Value: $$E[Y] = r(1-p)/p$$
Variance: $$Var[Y] = r(1-p)/p^2$$
Arena Generation: use convolution of geometric
Spreadsheet Generation: use convolution of geometric
Modeling: the number of failures prior to the $$r^{th}$$ success
Poisson Pois($$\lambda$$)
Parameters: $$\lambda > 0$$
PMF: $$P[X=x] = \frac{e^{-\lambda}\lambda^{x}}{x!} \quad x = 0, 1, \ldots$$
Inverse CDF: no closed form available
Expected Value: $$E[X] = \lambda$$
Variance: $$Var[X] = \lambda$$
Arena Generation: POIS($$\lambda$$[,Stream])
Spreadsheet Generation: not available, approximate with lookup table approach
Modeling: the number of occurrences during a period of time
Discrete Uniform DU($$a, b$$)
Parameters: $$a \leq b$$
PMF: $$P[X=x] = \frac{1}{b-a+1} \quad x = a, a+1, \ldots, b$$
Inverse CDF: $$F^{-1}(u) = a + \lfloor(b-a+1)u\rfloor$$
Expected Value: $$E[X] = (b+a)/2$$
Variance: $$Var[X] = \left( \left( b-a+1\right)^2 -1 \right)/12$$
Arena Generation: a + AINT((b-a+1)*UNIF(0,1))