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Collection by: Paul E Pfeiffer. E-mail the author

# Appendix C: Data on some common distributions

Module by: Paul E Pfeiffer. E-mail the author

Summary: Data are provided on some commonly used discrete and absolutely continuous distributions. Matlab procedures are provided for some.

## Discrete distributions

1. Indicator functions X = I E P ( X = 1 ) = P ( E ) = p P ( X = 0 ) = q = 1 - p X = I E P ( X = 1 ) = P ( E ) = p P ( X = 0 ) = q = 1 - p
E [ X ] = p Var [ X ] = p q M X ( s ) = q + p e s g X ( s ) = q + p s E [ X ] = p Var [ X ] = p q M X ( s ) = q + p e s g X ( s ) = q + p s
(1)
2. Simple random variable X = i = 1 n t i I A i (a primitive form) P ( A i ) = p i X = i = 1 n t i I A i (a primitive form) P ( A i ) = p i
E [ X ] = i = 1 n t i p i Var [ X ] = i = 1 n t i 2 p i q i - 2 i < j t i t j p i p j M X ( s ) = i = 1 n p i e s t i E [ X ] = i = 1 n t i p i Var [ X ] = i = 1 n t i 2 p i q i - 2 i < j t i t j p i p j M X ( s ) = i = 1 n p i e s t i
(2)
3. Binomial(n,p)(n,p)X=i=1nIEiwith{IEi:1in}iidP(Ei)=pX=i=1nIEiwith{IEi:1in}iidP(Ei)=p
P(X=k)=C(n,k)pkqn-kP(X=k)=C(n,k)pkqn-k
(3)
E[X]=np Var [X]=npqMX(s)=(q+pes)ngX(s)=(q+ps)nE[X]=np Var [X]=npqMX(s)=(q+pes)ngX(s)=(q+ps)n
(4)
MATLAB:          P(X=k)=ibinom(n,p,k)P(Xk)=cbinom(n,p,k)P(X=k)=ibinom(n,p,k)P(Xk)=cbinom(n,p,k)
4. Geometric(p)(p)P(X=k)=pqkk0P(X=k)=pqkk0
E[X]=q/p Var [X]=q/p2MX(s)=p1-qesgX(s)=p1-qsE[X]=q/p Var [X]=q/p2MX(s)=p1-qesgX(s)=p1-qs
(5)
If Y-1Y-1 geometric (p)(p), so that P(Y=k)=pqk-1k 1 P(Y=k)=pqk-1k 1 , then
E[Y]=1/p Var [X]=q/p2MY(s)=pes1-qesgY(s)=ps1-qsE[Y]=1/p Var [X]=q/p2MY(s)=pes1-qesgY(s)=ps1-qs
(6)
5. Negative binomial(m,p)(m,p). X is the number of failures before the mth success. P(X=k)=C(m+k-1,m-1)pmqkk0P(X=k)=C(m+k-1,m-1)pmqkk0.
E[X]=mq/p Var [X]=mq/p2MX(s)=p1-qesmgX(s)=p1-qsmE[X]=mq/p Var [X]=mq/p2MX(s)=p1-qesmgX(s)=p1-qsm
(7)
For Ym=Xm+mYm=Xm+m, the number of the trial on which mth success occurs. P(Y=k)=C(k-1,m-1)pmqk-mkmP(Y=k)=C(k-1,m-1)pmqk-mkm.
E[Y]=m/p Var [Y]=mq/p2MY(s)=pes1-qesmgY(s)=ps1-qsmE[Y]=m/p Var [Y]=mq/p2MY(s)=pes1-qesmgY(s)=ps1-qsm
(8)
MATLAB:          P(Y=k)=nbinom(m,p,k)P(Y=k)=nbinom(m,p,k)
6. Poisson(μ)(μ). P(X=k)=e-μμkk!k0P(X=k)=e-μμkk!k0
E[X]=μ Var [X]=μMX(s)=eμ(es-1)gX(s)=eμ(s-1)E[X]=μ Var [X]=μMX(s)=eμ(es-1)gX(s)=eμ(s-1)
(9)
MATLAB:          P(X=k)=ipoisson(m,k)P(Xk)=cpoisson(m, k)P(X=k)=ipoisson(m,k)P(Xk)=cpoisson(m, k)

## Absolutely continuous distributions

1. Uniform(a,b)(a,b)fX(t)=1b-aa<t<bfX(t)=1b-aa<t<b (zero elsewhere)
E[X]=b+a2 Var [X]=(b-a)212MX(s)=esb-esas(b-a)E[X]=b+a2 Var [X]=(b-a)212MX(s)=esb-esas(b-a)
(10)
2. Symmetric triangular ( - a , a ) ( - a , a ) f X ( t ) = ( a + t ) / a 2 - a t < 0 ( a - t ) / a 2 0 t a f X ( t ) = ( a + t ) / a 2 - a t < 0 ( a - t ) / a 2 0 t a
E [ X ] = 0 Var [ X ] = a 2 6 M X ( s ) = e a s + e - a s - 2 a 2 s 2 = e a s - 1 a s · 1 - e - a s a s E [ X ] = 0 Var [ X ] = a 2 6 M X ( s ) = e a s + e - a s - 2 a 2 s 2 = e a s - 1 a s · 1 - e - a s a s
(11)
3. Exponential ( λ ) ( λ ) f X ( t ) = λ e - λ t t 0 f X ( t ) = λ e - λ t t 0
E [ X ] = 1 λ Var [ X ] = 1 λ 2 M X ( s ) = λ λ - s E [ X ] = 1 λ Var [ X ] = 1 λ 2 M X ( s ) = λ λ - s
(12)
4. Gamma(α,λ)(α,λ)fX(t)=λαtα-1e-λtΓ(α)t0fX(t)=λαtα-1e-λtΓ(α)t0
E[X]=αλ Var [X]=αλ2MX(s)=λλ-sαE[X]=αλ Var [X]=αλ2MX(s)=λλ-sα
(13)
MATLAB:          P(Xt)=gammadbn(α,λ,t)P(Xt)=gammadbn(α,λ,t)
5. NormalN(μ,σ2)N(μ,σ2)fX(t)=1σ2πexp-12t-μσ2fX(t)=1σ2πexp-12t-μσ2
E[X]=μ Var [X]σ2MX(s)=expσ2s22+μsE[X]=μ Var [X]σ2MX(s)=expσ2s22+μs
(14)
MATLAB:          P(Xt)=gaussian(μ,σ2,t)P(Xt)=gaussian(μ,σ2,t)
6. Beta(r,s)(r,s)
fX(t)=Γ(r+s)Γ(r)Γ(s)tr-1(1-t)s-10<t<1,r>0,s>0fX(t)=Γ(r+s)Γ(r)Γ(s)tr-1(1-t)s-10<t<1,r>0,s>0
(15)
E[X]=rr+sVar[X]=rs(r+s)2(r+s+1)E[X]=rr+sVar[X]=rs(r+s)2(r+s+1)
(16)
MATLAB: fX(t)=beta(r,s,t)P(Xt)=betadbn(r,s,t)fX(t)=beta(r,s,t)P(Xt)=betadbn(r,s,t)
7. Weibull(α,λ,ν)(α,λ,ν)
FX(t)=1-e-λ(t-ν)α,α>0,λ>0,ν0,tνFX(t)=1-e-λ(t-ν)α,α>0,λ>0,ν0,tν
(17)
E[X]=1λ1/αΓ(1+1/α)+νVar[X]=1λ2/αΓ(1+2/λ)-Γ2(1+1/λ)E[X]=1λ1/αΓ(1+1/α)+νVar[X]=1λ2/αΓ(1+2/λ)-Γ2(1+1/λ)
(18)
MATLAB: (ν=0(ν=0 only)
fX(t)=weibull(a,l,t)P(Xt)=weibulld(a,l,t)fX(t)=weibull(a,l,t)P(Xt)=weibulld(a,l,t)
(19)

## Relationship between gamma and Poisson distributions

• If XX gamma (n,λ)(n,λ), then P(Xt)=P(Yn)P(Xt)=P(Yn) where YY Poisson (λt)(λt).
• If YY Poisson (λt)(λt), then P(Yn)=P(Xt)P(Yn)=P(Xt) where XX gamma (n,λ)(n,λ).

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