E.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: | \(Var[X] = p(1-p)\) |
| Generation: | BernoulliRV(probOfSuccess:Double, stream:RNStreamIf) |
| 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: | \(Var[X] = np(1-p)\) |
| Generation: | BinomialRV(pSuccess:Double, numTrials:Int, stream:RNStreamIf) |
| Spreadsheet Generation: | = BINOM.INV(n,p,RAND()) |
| 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: | \(Var[X] = (1-p)/p^2\) |
| Generation: | ShiftedGeometricRV(probOfSuccess:Double, stream:RNStreamIf) |
| 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: | \(Var[X] = r(1-p)/p^2\) |
| Generation: | NegativeBinomialRV(probOfSuccess:Double, numSuccess: Int, stream:RNStreamIf) |
| 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: | \(Var[Y] = r(1-p)/p^2\) |
| Generation: | value of NegativeBinomialRV minus 1 |
| 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: | \(Var[X] = \lambda\) |
| Generation: | PoissonRV(mean: Double, stream:RNStreamIfc) |
| 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: | \(Var[X] = \left( \left( b-a+1\right)^2 -1 \right)/12\) |
| Generation: | DUniformRV(min: Int, max:Int, stream:RNStreamIfc) |
| Spreadsheet Generation: | =RANDBETWEEN(a,b) |
| Modeling: | equal occurrence over a range of integers |