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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:1≤i≤n}iidP(Ei)=pX=∑i=1nIEiwith{IEi:1≤i≤n}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(X≥k)=cbinom(n,p,k)P(X=k)=ibinom(n,p,k)P(X≥k)=cbinom(n,p,k)
  4. Geometric(p)(p)P(X=k)=pqk∀k≥0P(X=k)=pqk∀k≥0
    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-1∼Y-1∼ geometric (p)(p), so that P(Y=k)=pqk-1∀k≥ 1 P(Y=k)=pqk-1∀k≥ 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)pmqk∀k≥0P(X=k)=C(m+k-1,m-1)pmqk∀k≥0.
    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-m∀k≥mP(Y=k)=C(k-1,m-1)pmqk-m∀k≥m.
    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!∀k≥0P(X=k)=e-μμkk!∀k≥0
    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(X≥k)=cpoisson(m, k)P(X=k)=ipoisson(m,k)P(X≥k)=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)t≥0fX(t)=λαtα-1e-λtΓ(α)t≥0
    E[X]=αλ Var [X]=αλ2MX(s)=λλ-sαE[X]=αλ Var [X]=αλ2MX(s)=λλ-sα
    (13)
    MATLAB:          P(X≤t)=gammadbn(α,λ,t)P(X≤t)=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(X≤t)=gaussian(μ,σ2,t)P(X≤t)=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(X≤t)=betadbn(r,s,t)fX(t)=beta(r,s,t)P(X≤t)=betadbn(r,s,t)
  7. Weibull(α,λ,ν)(α,λ,ν)
    FX(t)=1-e-λ(t-ν)α,α>0,λ>0,ν≥0,t≥νFX(t)=1-e-λ(t-ν)α,α>0,λ>0,ν≥0,t≥ν
    (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(X≤t)=weibulld(a,l,t)fX(t)=weibull(a,l,t)P(X≤t)=weibulld(a,l,t)
    (19)

Relationship between gamma and Poisson distributions

  • If X∼X∼ gamma (n,λ)(n,λ), then P(X≤t)=P(Y≥n)P(X≤t)=P(Y≥n) where Y∼Y∼ Poisson (λt)(λt).
  • If Y∼Y∼ Poisson (λt)(λt), then P(Y≥n)=P(X≤t)P(Y≥n)=P(X≤t) where X∼X∼ gamma (n,λ)(n,λ).

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