On the Beta Cumulative Distribution Function

Transcription

1 Alied Mathematical Scieces, Vol. 12, 218, o. 1, HIKARI Ltd, htts://doi.org/ /ams O the Beta Cumulative Distributio Fuctio Khaled M. Aludaat Deartmet of Statistics, Yarmouk Uiversity Irbid, Jorda Coyright 218 Khaled M. Aludaat. This article is distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. Abstract May authors have suggested differet aroximatios for the cumulative distributio fuctio of the beta distributio based o umerical calculatios by comuters. Such aroximatios require log ad hard rocesses to be achieved. I this aer we roose a ractical method for the calculatio of the beta cumulative distributio fuctio usig differetiatio istead of itegratio. Keywords: Beta distributio, egative biomial distributio 1. Itroductio The beta distributio is a cotiuous robability distributio defied i the uit iterval ad ossesses two ositive arameters α ad β who cotrol the shae of the distributio. The beta distributio is very imortat model for radom behavior of roortios. For examle, i Bayesia iferece, the beta distributio is used as a cojugate rior for biomial ad egative biomial robabilities. I sort scieces, i articular i the battig roblem of a layer, the best way to rereset the rior exectatio is the beta distributio. Sulaima et al. (1999) used the beta distributio to model the Malaysia sushie data for a te-year eriod. A radom variable X is said to have a beta distributio with arameters α > ad β >, deeoted, by X: Beta(α, β), the robability desity fuctio is give by f(x; α, β) = Γ(α+β) Γ(α)Γ(β) xα 1 (1 x) β 1, < x < 1. The cumulative distributio fuctio of X: Beta(α, β) is give by Γ(α + β) I (α, β) = P(X ) = Γ(α)Γ(β) xα 1 (1 x) β 1 dx (1)

2 462 Khaled M. Aludaat The beta distributio ejoys several ice roerties. 1. If α = β, the the shae of f(x; α, β) is symmetric, uimodal ad the Mode = mea = media = If α < β, the the shae of f(x; α, β) right-skewed, uimodal ad Mea > media > mode < If α > β, the the shae of f(x; α, β) left-skewed, uimodal ad Mea<media<mode> I (α, β) = 1 I 1 (β, α) 5. I (α, β) =, I 1 (α, β) = 1 ad I (1,1) = 6. I (α, β) = I (α 1, β) + (1 )I (α, β 1), for α, β = 1, 2, 3, 7. I (α, β) has a maximum value at = α 1, for α > 1 ad β > 1. α+β 2 8. I (α, β) if α > β ad I (α, β) if α < β. The had-calculatio itegral i (1) is difficult eve whe the values of α ad β are iteger, sice it requires reetitio of itegratio by arts. I this aer, we derive a formula to comute the value of (1) by had-calculatio whe the arameters values are itegers. 2. Relatio to the other distributios I this sectio, we reset some theorems that related the beta distributio to some imortat distributio such as the biomial, the egative biomial, the F- distributio ad the studet distributio. Most of these results ca be foud i Johso et al. (1992). Theorem 1. Let X: Beta(α, β), where α ad β are ositive itegers. Also let Y: Bi(α + β 1, ), a biomial distributio with arameters α + β 1 ad. The P(Y α) = I (α, β). Proof. By defiitio of I (α, β), Γ(α + β) I (α, β) = Γ(α)Γ(β) xα 1 (1 x) β 1 dx, = (α + β 1)(α + β 1)! (α 1)! (β 1)! x α 1 (1 x) β 1 dx, (α 1 + β 1)! = (α 1)! (β 1)! xα 1 (1 x) β 1 dx. If we itegrate (α 1) times i the above equatio, we get

3 O the beta cumulative distributio fuctio 463 where = α + β 1. α+β 1 I (α, β) = (α + β 1)! k! (α + β 1 k)! k (1 ) α+β 1 k, =! k! ( k)! k (1 ) k = P(Y α), Theorem 2. Let Y: Bi(α + β 1, ) ad T: NBi(α, ), a egative biomial distributio with arameters α ad. The P(Y α) = P(T α + β 1). Proof. Let = α + β 1. Sice P(Y α) =! k! ( k)! k (1 ) k, the! P(Y = k) = k! ( k)! k (1 ) k = P(T = ). k So P(T = ) = k P(Y = k). Hece k P(Y α) = k! k! ( k)! k (1 ) k, α+β 1 = Hece, P(Y α) = P(T ). ( 1)! = (k 1)! ( k)! k (1 ) k, (α + β 2)! (k 1)! (α + β 1 k)! k (1 ) α+β 1 k. This result meas that the biomial ad the egative biomial sequeces are iverse to each other, i.e., ay evet ca be exresses i terms of the two sequeces. The followig theorem is a straightforward geeralizatio of Theorem 1 ad Theorem 2. Theorem 3. Let X: Beta(α, β), Y: Bi(α + β 1, ) ad T: NBi(α, ). The P(X < ) = P(T α + β 1). Theorem 4. Let T has a studet radom variable with ν degrees of freedom. If T X = v+t 2, the X: Beta (ν, ν ) ad P(T < t) = I 2 2 t ( ν, ν ). 2 2

4 464 Khaled M. Aludaat Theorem 5. Let T has a studet radom variable with ν degrees of freedom. If X = ν 2, the v+t X: Beta ( ν, ν ) ad P(T < t) = I 2 2 t ( ν, ν ) ad A 2 2 (a ) = 1 I ν ν (ν, 1 ), where ν+a A ( a ) = P( a < T < a). the limitig values : A ν () = ad ν A ( ν ) = 1 Theorem 6. Let X: Beta(α, β) ad F has a Fisher distributio with ν 1 ad ν 2 degrees of freedom. The P(F f(v 1v v 2 )) = I ν2 ( ν, ν ). 2 2 ν2+ν1f Theorem 7. Let : Uif(,1), a uiform distributio, the U: Beta(1,1) ad U 2 : Beta ( 1 2, 1). Theorem 8. Let H be a radom variable distributed accordig to stadard arcsi over the iterval (,1). The H: Beta ( 1, 1 ), with cumulative distributio fuctio 2 2 F(h) = 2 π arcsi( h) = 1 π arcsi(2h 1) Recurrece relatio ad mai result From Theorem 3, we get ca relate the cumulative distributio fuctio of X: Beta(α, β) ad the cumulative distributio fuctio of T α : NBi(α, ) as follows P(X ) = P(T α α + β 1). 1. If α = 1, the T 1 has a geometric distributio with cumulative distributio fuctio F() = P(T 1 ) = 1 q = 1 q β = I (1, β), where q = 1 ad = α + β If α = 2, the ( 1)! F() = P(T 2 ) = ( 2)! 2 (1 ) 2, i=2 = 2 (1 + 2q + 3q ( 1)q 2 ). Let Q = q + q q 1 = q q 1 q So 2 F() = (2 1)! (2 1) Q q (2 1) = = 2 ( 1 (2 1)! q i=1 qi = Q q (1 + 2q + 3q2 + + ( 1)q 2 ) 2 (2 1)! ).= I (2, β), (2 1) q (q q (2 1) 1 q ),

5 O the beta cumulative distributio fuctio Similarly, if α = 3, the F() = 3 (3 1) Q (3 1)! q (3 1) = 3 (3 1) q (3 1)! q (3 1) (q2 1 q ), 4. For α = r, we have that = 3 2 (3 1)! q 1 ( qi 2 i=2 F() = r (r 1) Q (r 1)! q (r 1) = r (r 1) (qr 1 q (r 1)! q (r 1) 1 q ), = r (r 1) 1 ). ( qi) = I (r 1)! q (r 1) (r, β) i=r 1 Now the icomlete beta ca be evaluated via the followig equatio Hece I (α, β) = P(X ) = P(T α α + β 1) = F(), = α (α 1) (qα 1 q (α 1)! q (α 1) 1 q ), = α Γ(α + β) Γ(α)Γ(β) xα 1 (1 x) β 1 dx (α 1) α+β 2 (α 1)! q (α 1) ( = α i=a 1 (α 1) qi ). α+β 2 (α 1)! q (α 1) ( i=a 1 So we ca evaluate the robability P(k X h) via P(k X h) = I h (α, β) I k (α, β). qi ) (2) I the ext sectio, we give a umerical examle to illustrate the formula (2). 4. Numerical examles I this sectio we comute the I (α, β) for differet values of α ad β of x i the iterval [, 1] Table 1. Values of I (α, β) usig differetiatio formula x I x (2,2) I x (1,3) I x (2,5) I 1 x (5,2) I x (8,1)

6 466 Khaled M. Aludaat Coclusio The calculatio of the beta cumulative distributio of arameters α ad β, where α ad β iteger umbers, requires α 1 oeratios of itegratios by arts. Such oeratios are difficult to be achieved by had calculatio. Our method has a great advatage i chagig the itegratio by arts to a simle differetiatio of a fuctio related to the egative biomial cumulative distributio fuctio. The differetiatio is a easy rocess that saves time comared to the rocess of itegratio. Refereces [1] M. Abramowitz ad A. I. Stegu, Hadbook of Mathematical Fuctios with Formulas, Grahs, ad Mathematical Tables, Alied Mathematics Series, Vol. 55, Dover Publicatios, New York [2] N. L. Johso, S. Kotz ad A. N. Kem, Uivariate Discrete Distributios, 2 d editio, Joh Wiley & Sos, [3] A. Reyi, Probability Theory, North-Hollad, 197. [4] M. Y. Sulaima, W. M. Hlaig Oo, M. A. Wahab ad A. Zakaria, Alicatio of beta distributio model to Malaysia sushie data, Reewable Eergy, 18 (1999), htts://doi.org/1.116/s (99)2-6 [5] Wikiedia.org./wiki/Cumulative distributio-fuctio Received: March 1, 218; Published: Aril 1, 218