Saisical Disribuions 1 Discree Disribuions 1 The uniform disribuion A random variable (rv) X has a uniform disribuion on he n-elemen se A = {x 1,x 2,,x n } if P (X = x) =1/n whenever x is in he se A The following alies o he secial case A = {1, 2,,n} Suor: x =1, 2,,n Parameers: n Probabiliy funcion (f): f(x n) =1/n, x =1, 2,, n Momen generaing funcion (mgf): M() = e (1 e n ) n(1 e ), 0, and M() = 1 for =0 Mean and variance: E(X) =(n +1)/2, Var(X) =(n 2 1)/12 2 The Bernoulli disribuion A random variable X has a Bernoulli disribuion wih arameer (0 1) if i akes only values 0 and 1 wih robabiliies 1 and, resecively Noaion: Bernoulli() Suor: x =0, 1 Parameers: 0 1 Probabiliy funcion (f): f(x ) = x (1 ) 1 x for x =0, 1 Momen generaing funcion (mgf): M() =e +(1 ) Mean and variance: E(X) =, Var(X) = (1 ) 3 The binomial disribuion If X 1,,X n are indeenden, idenically disribued (iid) Bernoulli rv s wih arameer, hen X = X 1 + + X n has a binomial disribuion wih arameers n and If a lo conains N iems, of which A are defecive, and if a samle of size n iems is chosen wih relacemen from he lo, hen he number of defecive iems in he samle, X, has a binomial disribuion wih arameers n and = A/N Noaion: Bin(n, ) Suor: x =0,,n Parameers: 0 1 and n {1, 2,} Pf: f(x n, ) = n! x!(n x)! x (1 ) n x Mgf: M() =[e +(1 )] n Mean and variance: E(X) = n, Var(X) = n(1 ) 4 The hyergeomeric disribuion If a lo conains N iems, of which A are defecive (and B = N A are non-defecive), and if a samle of size n iems is chosen wihou relacemen from he lo, hen he number of defecive iems in he samle, X, has a hyergeomeric disribuion wih arameers A, B, and n Noaion: Hy(A,B,n) 1
Suor: max{0, n B} x min{n, A} Parameers: A, B, n {1, 2,} Pf: f(x A,B,n)= A! B! Mean and variance: E(X) =n, x!(a x)! (n x)!(b n+x)! / (A+B)! n!(a+b n)! Var(X) =n(1 ) A+B n A+B 1, where = A A+B 5 The geomeric disribuion Consider a sequence of Bernoulli rials in which he oucome of any rial is eiher 1 (success) or 0 (failure) and he robabiliy of success on any rial is Le X denoe he number of rials needed so ha he success occurs for he firs ime a he las rial Then, X is said o have a geomeric disribuion wih arameer Noaion: Geo() Suor: osiive inegers Parameers: 0 1 Pf: f(x ) =(1 ) x 1, x =1, 2, 3, e Mgf: M() = 1 (1 )e, <ln(1/(1 )) Mean and variance: E(X) = 1, 1 Var(X) = 2 Noe An alernaive definiion of geomeric disribuion: number of failures, Y, ha occur before he firs success is obained (his is he definiion adoed in he ex) Since X 1 =Y, he formulas for Y are easily derived from hose for X: Suor: non-negaive inegers Parameers: 0 1 Pf: f(x ) =(1 ) x, x =0, 1, 2, Mgf: M() = 1 (1 )e, <ln(1/(1 )) Mean and variance: E(X) = 1 1, Var(X) = 2 6 The negaive binomial disribuion Consider a sequence of Bernoulli rials in which he oucome of any rial is eiher 1 (success) or 0 (failure) and he robabiliy of success on any rial is Le X denoe he number of rials needed so ha he success occurs for he rh ime a he las rial Then, X is said o have a negaive binomial disribuion wih arameers r and If X 1,,X r are iid rv s and each has a Geo() disribuion, hen X = X 1 + + X r has a negaive binomial disribuion wih arameers r and Noaion: NB(r, ) Suor: x {r, r +1,r+2,} Parameers: r {1, 2, 3,} and 0 1 (x 1)! Pf: f(x r, ) = (r 1)!(x r)! r (1 ) x r [ ] e r, Mgf: M() = 1 (1 )e <ln(1/(1 )) Mean and variance: E(X) =r/, Var(X) = r(1 ) 2 2
Noe An alernaive definiion of negaive binomial disribuion: number of failures, Y, ha occur before he rh success is obained (his definiion is adoed in our ex) Since X r = Y, he formulas for Y are easily derived from hose for X: Suor: non-negaive inegers Parameers: r {1, 2, 3,} and 0 1 Pf: f(x r, ) = (r+x 1)! (r 1)!x! r (1 ) x, x =0, 1, 2, [ ] r, Mgf: M() = 1 (1 )e <ln(1/(1 )) Mean and variance: E(X) = r(1 ), Var(X) = r(1 ) 2 7 The Poisson disribuion Noaion: P oisson(λ) Suor: x {0, 1, 2,} Parameers: λ>0 Pf: f(x λ) = e λ λ x x! Mgf: M() =e λ(e 1) Mean and variance: E(X) =λ, Var(X) =λ 8 The mulinomial disribuion This is a mulivariae disribuion where each of n indeenden rials can resul in one of r yes of oucomes, and on each rial he robabiliies of he r oucomes are 1, 2,, r The variable here is a vecor (X 1,,X r ), where X i is he number of oucomes of ye i Suor: x i {0, 1, 2,n}, i =1, 2,r, x 1 + + x r = n Parameers: n, r - osiive inegers, n>r>1; 1, 2,, r - osiive numbers ha add u o one Pf: f(x 1,,x r )= n! x 1!x r! x 1 1 xr r Mgf: Mean and variance: E(X i )=n i, Var(X i )=n i (1 i ) 2 Coninuous disribuions 1 The uniform disribuion Noaion: U(a, b) Suor: x [a, b] Parameers: <a<b< Probabiliy densiy funcion (df): f(x a, b) = 1 b a Mgf: M() = eb e a (b a) Mean and variance: E(X) = a+b (b a)2 2, Var(X) = 12 3
Mode: a+b Median: 2 2 The exonenial disribuion Noaion: Ex(β) Suor: x 0 Parameers: β>0 Pdf: f(x β) =βe βx Mgf: M() = β β, <β Mean and variance: E(X) =1/β, Var(X) =1/β 2 Mode: 0 Median: ln2/β 3 The gamma disribuion For any α>0, he value Γ(α) denoes he inegral 0 x α 1 e x dx, and is called he gamma funcion Some roeries of he gamma funcion include: Γ(α+1) = αγ(α), Γ(n) =(n 1)! for any ineger n 1, and Γ(1/2) = π 1/2 Noaion: G(α, β) Suor: x 0 Parameers: α, β > 0 Pdf: f(x α, β) = βα Γ(α) xα 1 e βx ( Mgf: M() = β α, β ) <β Mean and variance: E(X) =α/β, Var(X) =α/β 2 Mode: (α 1)/β if α>1 and 0 for 0 <α 1 Noes Firs, noe ha Ex(β) = G(1,β) Nex, i follows from he formula of gamma mgf, ha if X i G(α i,β) are indeenden, hen X = X 1 + +X n has a gamma G(α 1 + +α n,β) disribuion In aricular, if X i Ex(β) are indeenden, hen X = X 1 + + X n G(n, β) 4 The chi-square disribuion wih n degrees of freedom Secial case of gamma disribuion G(α, β) wih α = n/2, β =1/2, n =1, 2, 3, Noaion: χ 2 n Suor: x 0 Parameers: n - a osiive ineger called he degrees of freedom 1 Pdf: f(x n) = x n/2 1 e 1 Γ(n/2)2 n/2 2 x ( ) Mgf: M() = 1 n/2, 1 2 <1/2 Mean and variance: E(X) =n, Var(X) =2n Mode: n 2ifn 2 and 0 for n =1 4
Noes The chi-square disribuion arises in connecion wih random samles from he normal disribuion: If Z 1,Z n are iid sandard normal variables, hen he quaniy X = Z 2 1 + + Z 2 n has he χ 2 n disribuion Consequenly, if X 1,X n are iid normal N(µ, σ) variables, hen he quaniy X = 1 σ 2 n (X i µ) 2 has he χ 2 n disribuion I is also rue ha under he above condiions he quaniy X = 1 σ 2 n (X i X) 2 has he χ 2 n 1 disribuion, where X is he samle mean of he X i s 5 The disribuion (Suden s disribuion) wih n degrees of freedom Noaion: n Suor: <x< Parameers: n - a osiive ineger called he degrees of freedom (df) Pdf: f(x n) = Γ[(n+1)/2] nπγ(n/2) (1 + x 2 /n) (n+1)/2 Mgf: Mean and variance: E(X) = 0 (for n>1), Var(X) =n/(n 2) (for n>2) Mode: 0 Median: 0 Noes The disribuion is relaed o he normal and chi-square disribuions, and arises in connecion wih random samles from he normal disribuion If Z is sandard normal and U is chi-square wih n df, hen he quaniy X = Z/ U/n has he disribuion wih n df Consequenly, if X 1,,X n are iid normal N(µ, σ) variables, hen he quaniy where X is he samle mean of he X i s and S 2 = 1 n 1 X = X µ S/ n, n (X i X) 2 is he samle variance of he X i s, has he n 1 disribuion 6 The F disribuion wih m and n degrees of freedom Noaion: F m,n Suor: x 0 5
Parameers: m and n - osiive inegers called he degrees of freedom (df) Pdf: f(x m, n) = Γ[(m+n)/2] Γ(m/2)Γ(n/2) (m/n)m/2 x m/2 1 (1 + mx/n) (m+n)/2 Mean and variance: E(X) = n n 2 (for n>2), Var(X) = 2n2 (m+n 2) m(n 2) 2 (n 4) (for n>4) Noes The F disribuion is relaed o chi-square disribuion, and arises in connecion wih wo indeenden random samles from normal disribuions If U and V are indeenden chi-square random variables wih m and n degrees of freedom, resecively, hen he quaniy X = U/m V/n has he F disribuion wih m and n df Consequenly, if X 1,,X m are iid normal N(µ 1,σ) variables, and if Y 1,,Y n are iid normal N(µ 2,σ) variables, hen he quaniy X = S 2 1 /S2 2 has he F m 1,n 1 disribuion, where S 2 1 = 1 m 1 are he samle variances of he X i s and he Y i s m (X i X) 2 and S2 2 = 1 n (Y i Y ) 2 n 1 7 The Bea disribuion Noaion: Bea(α, β) Suor: 0 x 1 Parameers: α, β > 0 Pdf: f(x α, β) = Γ(α+β) Γ(α)Γ(β) xα 1 (1 x) β 1 Mean and variance: E(X) = α α+β, Var(X) = αβ (α+β) 2 (α+β+1) Noe Secial case: Bea(1, 1) = U(0, 1) 8 The normal disribuion Noaion: N(µ, σ) Suor: <x< Parameers: <µ< and σ>0 Pdf: f(x µ, σ) = 1 2πσ e (x µ)2 2σ 2 Mgf: M() =e µ+σ2 2 /2 Mean and variance: E(X) =µ, Var(X) =σ 2 Mode: µ Median: µ 9 The lognormal disribuion Noaion: LN(µ, σ) 6
Suor: x 0 Parameers: <µ< and σ>0 Pdf: f(x µ, σ) = 1 (ln x µ)2 2πσx e 2σ 2 Mean and variance: E(X) =e µ+σ2 /2, Var(X) =(e σ2 1)e 2µ+σ2 Mode: e µ σ2 /2 Median: e µ Noe IfY is normal N(µ, σ) hen X = e Y is LN(µ, σ) 10 The Pareo disribuion Noaion: P (k, α) Suor: x k Parameers: k>0 and α>0 Pdf: f(x k, α) =αk α x (α+1) Mean and variance: E(X) = αk α 1 (for α>1), Var(X) = αk 2 (α 2)(α 1) 2 (for α>2) Mode: k Median: 2 1/α k 11 The Lalace (double exonenial) disribuion Noaion: L(µ, σ) Suor: <x< Parameers: <µ< and σ>0 Pdf: f(x µ, σ) = 1 x µ 2σ e σ Mgf: M() = eµ, 1/σ < < 1/σ 1 2 σ 2 Mean and variance: E(X) =µ, Var(X) =2σ 2 Mode: µ Median: µ 12 The Cauchy disribuion Noaion: Suor: <x< Parameers: <µ< and σ>0 [ Pdf: f(x µ, σ) =(πσ) 1 1+{(x θ)/σ)} 2] 1 Mean and variance: Do no exis Mode: µ Median: µ Noe The sandard Cauchy disribuion corresonds o µ = 0 and σ =1 7
13 The Weibull disribuion Noaion: W (α, β) Suor: x 0 Parameers: α, β > 0 Pdf: f(x α, β) = β ( x ) β 1 α α e (x/α)β Mean and variance: E(X) =αγ(1 + 1/β), Var(X) =α 2 [Γ(1 + 2/β) {Γ(1 + 1/β)} 2 ] ) 1/β Mode: α if β>1 and 0 for 0 <β 1 ( β 1 β Median: α(ln 2) 1/β Noes IfX is sandard exonenial, hen W = αx 1/β has he W (α, β) disribuion 14 The bivariae normal disribuion Join coninuous disribuion of X and Y Noaion: N(µ 1,µ 2,σ 1,σ 2,ρ) Suor: <x,y< Parameers: <µ 1,µ 2 <, σ 1,σ 2 > 0, 1 <ρ<1 Pdf: Mean and variance: 1 f(x, y) = 2πσ 1 σ 2 1 ρ 2 e 1 2(1 ρ 2 ) [ (x µ 1 ) 2 σ 2 1 ] + (y µ 2 )2 σ 2 2 2ρ(x µ 1 )(y µ 2 ) σ 1 σ 2 E(X) =µ 1, E(Y )=µ 2, Var(X) =σ 2 1, Var(Y )=σ 2 2, Cov(X, Y )=σ 1 σ 2 ρ 8