Appendix D — Distributions

D.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

D.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:	\(V[X] = \frac{(b-a)^2}{12}\)
Generation:	UniformRV(min: Double, max:Double, stream:RNStreamIfc)
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:	\(Var[X] = \sigma^2\)
Generation:	NormalRV(mean:Double, variance:Double, stream:RNStreamIfc)
Spreadsheet Generation:	= NORM.INV(RAND(), \(\mu\), \(\sigma\))
Modeling:	task times, errors

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:	\(Var[X] = 1/\lambda^2\)
Generation:	ExponentialRV(mean:Double, stream:RNStreamIfc)
Spreadsheet Generation:	= \((-1/\lambda)\)LN(1-RAND())
Modeling:	time between arrivals, time to failure
	highly variable task time

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:	\(Var[X] = \left(\dfrac{\beta^2}{\alpha}\right)\biggl\lbrace 2\Gamma\left(\dfrac{2}{\alpha}\right) - \left(\dfrac{1}{\alpha}\right)\biggl(\Gamma\left(\dfrac{1}{\alpha}\right)\biggr)^2\biggr\rbrace\)
Generation:	WeibullRV(shape:Double, scale:Double, stream:RNStreamIfc)
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:	\(Var[X] = r\beta^2\)
Generation:	use GammaRV class
Spreadsheet Generation:	= GAMMA.INV(RAND(), \(r\), \(\beta\))
Modeling:	task times, lead time, time to failure,

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:	\(Var[X] = \alpha \beta^2\)
Generation:	GammaRV(shape:Double, scale:Double, stream:RNStreamIfc)
Spreadsheet Generation:	= GAMMA.INV(RAND(), \(\alpha\), \(\beta\))
Modeling:	task times, lead time, time to failure,

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:	\(Var[X] = \dfrac{\alpha_1\alpha_2}{(\alpha_1 + \alpha_2)^2(\alpha_1 + \alpha_2+1)}\)
Generation:	BetaRV(alpha1:Double, alpha2:Double, stream:RNStreamIfc)
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:	\(Var[X] = \sigma_{l}^{2} = e^{2\mu + \sigma^{2}}\left(e^{\sigma^{2}} - 1\right)\)
Generation:	LognormalRV(mean:Double, variance:Double, stream:RNStreamIfc)
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:	\(Var[X] = \dfrac{a^2 + b^2 + m^2 -ab -am -bm}{18}\)
Generation:	TriangularRV(min:Double, mode:Double, max:Double, stream:RNStreamIfc)
Spreadsheet Generation:	implement \(F^{-1}(u)\) as VBA function
Modeling:	task times, activity time when data is limited