Analytical valuation of some non-elementary integrals involving some exponential, hyperbolic and trigonometric elementary functions and derivation of new probability measures generalizing the gamma-type and normal distributions
aa r X i v : . [ m a t h . G M ] J un Analytical evaluation of some non-elementaryintegrals involving some exponential, hyperbolic andtrigonometric elementary functions and derivationof new probability measures generalizing thegamma-type and normal distributions
Victor Nijimbere
Abstract.
The non-elementary integrals involving elementary exponential, hyperbolic andtrigonometric functions, R x α e ηx β dx, R x α cosh (cid:0) ηx β (cid:1) dx, R x α sinh (cid:0) ηx β (cid:1) dx, R x α cos (cid:0) ηx β (cid:1) dx and R x α sin (cid:0) ηx β (cid:1) dx where α, η and β are real or complex constants are evaluated interms of the confluent hypergeometric function F and the hypergeometric function F .The hyperbolic and Euler identities are used to derive some identities involving exponen-tial, hyperbolic, trigonometric functions and the hypergeometric functions F and F .Having evaluated, these non-elementary integrals, some new probability measures gener-alizing the gamma-type and Gaussian distributions are also obtained. The obtained gen-eralized distributions may, for example, allow to perform better statistical tests than thosealready known (e.g. chi-square ( χ ) statistical tests and other statistical tests constructedbased on the central limit theorem (CLT)). Keywords.
Non-elementary integrals, Hypergeometric function, Confluenthypergeometric function, Probability measure, Gamma-type distributions, Gaussian-typedistributions.
The confluent hypergeometric function F and the hypergeometric function F are used throughout this paper. There are defined here for reference. Definition 1.1.
The confluent hypergeometric function, denoted as F , is a specialfunction given by the series [1, 12] F ( a ; b ; x ) = ∞ X n = ( a ) n ( b ) n x n n ! , (1) V. Nijimbere where a and b are arbitrary constants, ( ϑ ) n = Γ ( ϑ + n ) / Γ ( ϑ ) (Pochhammer’snotation [1]) for any complex ϑ , with ( ϑ ) =
1, and Γ is the standard gammafunction [1, 12]. Definition 1.2.
The hypergeometric function F , is a special function given bythe series [1, 12] F ( a ; b, c ; x ) = ∞ X n = ( a ) n ( b ) n ( c ) n x n n ! , (2)where a, b and c are arbitrary constants, ( ϑ ) n = Γ ( ϑ + n ) / Γ ( ϑ ) (see definition1.2)[1, 12]. Definition 1.3.
An elementary function is a function of one variable constructedusing that variable and constants, and by performing a finite number of repeatedalgebraic operations involving exponentials and logarithms. An indefinite integralwhich can be expressed in terms of elementary functions is an elementary integral.And if, on the other hand, it cannot be evaluated in terms of elementary functions,then it is non-elementary [6, 15].One of the goals of this work is to show how non-elementary integrals havingone of the types Z x α e ηx β dx, Z x α cosh (cid:16) ηx β (cid:17) dx, Z x α sinh (cid:16) ηx β (cid:17) dx, (3) Z x α cos (cid:16) ηx β (cid:17) dx and Z x α sin (cid:16) ηx β (cid:17) dx, (4)where α, η and β are real or complex constants can be evaluated in terms of thespecial functions F and F . These integrals are the generalization of the non-elementary integrals evaluated by Nijimbere [7–9], and have been not evaluatedbefore. For instance, if α <
0, the integrals in (4) become, respectively, the (in-definite) sine and cosine integrals which are evaluated in Nijimbere [8, 9]. If, onthe other hand, α =
0, the non-elementary integrals in (3) and (4) reduce to thenon-elementary integrals evaluated in Nijimbere [7].However, it is important to observe that the integrals in (3) and (4) may beelementary or non-elementary depending on the values of the constants α and β .If, for instance, α = β −
1, then the integral Z x α e ηx β dx = ηβ Z ηβx β − e ηx β dx = e ηx β ηβ + C (5) ome non-elementary integrals and applications in probability theory e ηx β . Inthat case, the other integrals in (3) and (4) are also elementary since they can beexpressed as linear combination of integrals such that in (5) using the hyperbolicidentitiescosh (cid:16) ηx β (cid:17) = (cid:16) e ηx β + e − ηx β (cid:17) / , sinh (cid:16) ηx β (cid:17) = (cid:16) e ηx β − e − ηx β (cid:17) / (cid:16) ηx β (cid:17) = (cid:16) e iηx β + e − iηx β (cid:17) / , sin (cid:16) ηx β (cid:17) = (cid:16) e iηx β − e − iηx β (cid:17) / ( i ) . Using Liouville 1835’s theorem, it can readily be shown that if α is not an inte-ger and α = β −
1, then the integrals in (4) and (5) are non-elementary [6, 15].Another goal of this work is to obtain some identities (or formula) involving ex-ponential, hyperbolic, trigonometric functions and the hypergeometric functions F and F using the Euler and hyperbolic identities. Other interesting identitiesinvolving hypergeometric functions may be found, for example, in [2, 5, 7–9, 14].Non-elementary Integrals with integrands involving generalized hypergeometricfunctions and identities of generalized hypergeometric series have also been ex-amined in Nijimbere [11].It is well known that numerical integrations (or approximations) are expensiveand their main drawback is that they are associated with computational errorswhich become very large as the integration limits become large. Thus, the ana-lytical method used in this paper (developed by Nijimbere [9]) is very importantin order to avoid computational methods, see for example the case of Dawson’sintegral and related functions in mathematical physics [10].Using the fact that g ( x ) = e − ηx β , x ∈ R , η ∈ R + , is in the L p -space, p > β ∈ R , some finite measure, µ ( {−∞ , x } ) < ∞ , can be defined for all x ∈ R . Moreover, if X = h ( x ) , x ∈ R is some random variable, h : R → R issome well-defined function (e.g. h ( x ) = x ), then it is possible to define probabilitymeasures in terms of the Lebesgue measure dx as µ ( dx ) = A g ( x ) dx, x ∈ Ω , and Ω ⊆ R , satisfying the integrability condition R Ω ⊆ R | X | α µ ( dx ) < ∞ , α = , α > − β − A being a (normalization) constant. In that case, probability measures(or distributions) that generalize the gamma-type and Gaussian-type distributionsmay be constructed, and corresponding distribution functions and moments can beevaluated as well. For example, it can be shown using the results in this paper thatthe n th moments of the Gaussian random variable are given by the formula M ( X n ) = θ n √ π n X l = Γ ( l + / ) C n l (cid:18) σ θ (cid:19) l , l ≤ n and ( l ) ∈ N , V. Nijimbere where θ ∈ R is the mean of the Gaussian random variable and σ > F and F areobtained. In section 3, probability measures that generalize the gamma-type andGaussian-type distributions are constructed, and their corresponding distributionfunctions are written in terms of the confluent hypergeometric function. Formulasto evaluate the n th moments are also derived in section 3. A general discussion isgiven in section 4. Let first prove an important lemma which will be used throughout the paper.
Lemma 2.1.
Let j ≥ and m ≥ be integers, and let α, β and γ be arbitrarilyconstants. (i) Then j Y m = ( α + mβ + ) = ( α + ) β j (cid:18) α + β + (cid:19) j , (6)(ii) j Y m = ( α + mβ + )= ( α + )( α + β + )( β ) j (cid:18) α + β + β + (cid:19) j (cid:18) α + β + β + (cid:19) j (7)(iii) and j + Y m = ( α + mβ + )= ( α + )( α + β + )( β ) j (cid:18) α + β + β + (cid:19) j (cid:18) α + β + β + (cid:19) j . (8) ome non-elementary integrals and applications in probability theory Proof. (i) Making use of Pochhammer’s notation [1,12], see definition 1.3 yields j Y m = ( α + mβ + ) = ( α + ) j Y m = ( α + mβ + )= ( α + ) β j j Y m = (cid:18) α + β + m (cid:19) = ( α + ) β j j Y m = (cid:18) α + β + + m − (cid:19) = ( α + ) β j (cid:18) α + β + (cid:19) j . (9)(ii) Observe that j Y m = ( α + mβ + ) = j − Y l = ( α + l ( β ) + β + ) j Y l = ( α + l ( β ) + ) . (10)Then, making use of Pochhammer’s notation as before gives j − Y l = ( α + l ( β ) + β + ) = ( α + β + )( β ) j (cid:18) α + β + β + (cid:19) j (11)and j Y l = ( α + l ( β ) + ) = ( α + )( β ) j (cid:18) α + β + β + (cid:19) j . (12)Hence, multiplying (11) with (12) gives (7).(iii) Observe that j + Y m = ( α + mβ + ) = j Y l = ( α + l ( β ) + ) j Y l = ( α + l ( β ) + β + ) . (13)Once again, using again Pochhammer’s notation yields j Y l = ( α + l ( β ) + β + ) = ( α + β + )( β ) j (cid:18) α + β + β + (cid:19) j . (14)Hence, multiplying (14) with (12) gives (8). V. Nijimbere R x α e ηx β dx, R x α cosh (cid:0) ηx β (cid:1) dx, R x α sinh (cid:0) ηx β (cid:1) dx Proposition 2.2.
Let η and β be nonzero constants ( η = , β = ), and α be anyconstant different from − ( α = − ). Then, Z x α e ηx β dx = x α + e ηx β α + F (cid:16) α + β + β ; − ηx β (cid:17) + C. (15) Proof.
The substitution u β = ηx β and (1) yields Z x α e ηx β dx = η α + β Z u α e u β du. (16)Performing successive integration by parts that increases the power of u gives Z u α e u β du = u α + e u β α + β + − βu α + β + e u β ( α + )( α + β + )+ β u α + β + e u β ( α + )( α + β + )( α + β + ) − β u α + β + e u β ( α + )( α + β + )( α + β + )( α + β + )+ · · · · + ( − ) j β j u α + jβ + e u β j Q m = ( α + mβ + ) + · · ·· = ∞ X j = ( − ) j β j u α + jβ + e u β j Q m = ( α + mβ + ) + C. (17)Using (6) in Lemma 2.1 yields Z u α e u β du = u α + e u β ∞ X j = ( − βu β ) jj Q m = ( α + mβ + ) + C = u α + e u β ∞ X j = ( − βu β ) j ( α + ) β j (cid:16) α + β + (cid:17) j + C ome non-elementary integrals and applications in probability theory = u α + e u β α + ∞ X j = ( − u β ) j (cid:16) α + β + (cid:17) j + C = u α + e u β α + ∞ X j = ( ) j ( − u β ) j (cid:16) α + β + (cid:17) j j ! + C = u α + e u β α + F (cid:16) α + β + β ; − u β (cid:17) + C. (18)Hence, using the fact u β = ηx β gives (7).Having evaluated (15), the following results hold. Theorem 2.3.
Let α be an arbitrarily real or complex constant, β a nonzero realor complex constant ( β = ), and η a nonzero real or complex constant with apositive real part (Re ( η ) > ). (i) Then, + ∞ Z x α e − ηx β dx = Γ (cid:16) α + β + β (cid:17) ( α + ) η α + β , (19) α > − β − , α = − if { α, β } ∈ R . (ii) Moreover, if the integrand is even, then + ∞ Z −∞ x α e − ηx β dx = Γ (cid:16) α + β + β (cid:17) ( α + ) η α + β . (20) Proof.
It can readily be shown using Proposition 2.2 and the asymptotic expansionof the confluent hypergeometric function (formula 13.1.5 in [1]) that + ∞ Z x α e − ηx β dx = lim x →∞ x α + e − ηx β α + F (cid:16) α + β + β ; ηx β (cid:17) = Γ (cid:16) α + β + β (cid:17) ( α + ) η α + β . (21)If the integrand is even, then R + ∞−∞ x α e − ηx β dx = R + ∞ x α e − ηx β dx , and thisgives (20).Theorem 2.3 is, for instance, the generalization of the Mellin transform of thefunction e − ηx β , Re { η } > , β >
0, where s = α + s = α + <
0, and the constant β can be negativeas well ( β < V. Nijimbere
As it will shortly be shown (see section 3), Theorem 2.3 can be used to obtainnew probability distributions that generalize the gamma-type and Gaussian-typedistributions that may lead to better statistical tests than those already known whichare based on the central limit theorem (CLT) [3].
Proposition 2.4.
Let η and β be nonzero constants ( η = , β = ), α be someconstant different from − ( α = − ) and α = − β − . Then, Z x α cosh (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i + C. (22) Proof.
The change of variable u β = ηx β yields Z x α cosh (cid:16) ηx β (cid:17) dx = η α + β Z u α cosh (cid:16) u β (cid:17) du + C. (23)Successive integration by parts that increases the power of u gives Z u α cosh (cid:16) u β (cid:17) du = u α + cosh (cid:0) u β (cid:1) α + β + − βu α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + )+ β u α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + )( α + β + ) − β u α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + )( α + β + )( α + β + )+ · · + β j u α + jβ + cosh (cid:0) u β (cid:1) j Q m = ( α + mβ + ) + · · · − β j + u α +( j + ) β + sinh (cid:0) u β (cid:1) j + Q m = ( α + mβ + ) − ·· = cosh (cid:16) u β (cid:17) ∞ X j = β j u α + jβ + j Q m = ( α + mβ + ) − sinh (cid:16) u β (cid:17) ∞ X j = β j + u α +( j + ) β + j + Q m = ( α + mβ + ) + C. (24)Using (7) and (8) in Lemma 2.1 yields ome non-elementary integrals and applications in probability theory Z u α cosh (cid:16) u β (cid:17) du = u α + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j − βu α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j + C = u α + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = ( ) j (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j j ! − βu α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = ( ) j (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j j ! + C = u α + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) F (cid:16) α + β + β , α + β + β ; u β (cid:17) − βu α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) F (cid:16) α + β + β , α + β + β ; u β (cid:17) + C. (25)Hence, using the fact u β = ηx β and rearranging terms gives (22). Proposition 2.5.
Let η and β be nonzero constants ( η = , β = ), α be someconstant different from − ( α = − ) and α = − β − . Then, Z x α sinh (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i + C. (26) Proof.
Making the change of variable u β = ηx β as before yields Z x α sinh (cid:16) ηx β (cid:17) dx = η α + β Z u α sinh (cid:16) u β (cid:17) du + C. (27)Performing successive integration by parts that increase the power of u as beforegives0 V. Nijimbere Z u α sinh (cid:16) u β (cid:17) du = u α + sinh (cid:0) u β (cid:1) α + β + − βu α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + )+ β u α + β + sinh (cid:0) u β (cid:1) ( α + )( α + β + )( α + β + ) − β u α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + )( α + β + )( α + β + )+ · · + β j u α + jβ + sinh (cid:0) u β (cid:1) j Q m = ( α + mβ + ) + · · − β j + u α +( j + ) β + cosh (cid:0) u β (cid:1) j + Q m = ( α + mβ + ) − ·· = sinh (cid:16) u β (cid:17) ∞ X j = β j u α + jβ + j Q m = ( α + mβ + ) − cosh (cid:16) u β (cid:17) ∞ X j = β j + u α +( j + ) β + j + Q m = ( α + mβ + ) + C. (28)Using (7) and (8) in Lemma 2.1 yields Z u α sinh (cid:16) u β (cid:17) du = u α + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j − βu α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j + C = u α + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = ( ) j (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j j ! − βu α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) ∞ X j = ( ) j (cid:16) u β (cid:17) j (cid:16) α + β + β (cid:17) j (cid:16) α + β + β (cid:17) j j ! + C = u α + sinh (cid:0) u β (cid:1) ( α + )( α + β + ) F (cid:16) α + β + β , α + β + β ; u β (cid:17) − βu α + β + cosh (cid:0) u β (cid:1) ( α + )( α + β + ) F (cid:16) α + β + β , α + β + β ; u β (cid:17) + C. (29)Hence, using the fact u β = ηx β and rearranging terms gives (26). ome non-elementary integrals and applications in probability theory Theorem 2.6.
For any constants α, β and η , α + β + h cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i = (cid:20) e ηx β F (cid:16) α + β + β ; − ηx β (cid:17) + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17)(cid:21) . (30) Proof.
Using the hyperbolic identity cosh (cid:0) ηx β (cid:1) = (cid:16) e ηx β + e − ηx β (cid:17) / Z x α cosh (cid:16) ηx β (cid:17) dx = (cid:18)Z x α e ηx β dx + Z x α e − ηx β dx (cid:19) = x α + ( α + ) h e ηx β F (cid:16) α + β + β ; − ηx β (cid:17) + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17)i + C. (31)Hence, Comparing (31) with (22) gives (30). Theorem 2.7.
For any constants α, β and η , α + β + h sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i = (cid:20) e ηx β F (cid:16) α + β + β ; − ηx β (cid:17) − e − ηx β F (cid:16) α + β + β ; ηx β (cid:17)(cid:21) . (32) Proof.
Using the hyperbolic identity sinh (cid:0) ηx β (cid:1) = (cid:16) e ηx β − e − ηx β (cid:17) / V. Nijimbere Z x α sinh (cid:16) ηx β (cid:17) dx = (cid:18)Z x α e ηx β dx − Z x α e − ηx β dx (cid:19) = x α + ( α + ) h e ηx β F (cid:16) α + β + β ; − ηx β (cid:17) − e − ηx β F (cid:16) α + β + β ; ηx β (cid:17)i + C. (33)Hence, Comparing (33) with (26) gives (32). Theorem 2.8.
For any constants α, β and η , e ηx β F (cid:16) α + β + β ; − ηx β (cid:17) = α + β + h sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) + cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i . (34) Proof.
The hyperbolic relation e ηx β = cosh (cid:0) ηx β (cid:1) + sinh (cid:0) ηx β (cid:1) and Propositions2.4 and 2.5 gives Z x α e ηx β dx = Z x α cosh (cid:16) ηx β (cid:17) dx + Z x α sinh (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i + x α + ( α + )( α + β + ) h sinh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17) − βηx β cosh (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; η x β (cid:17)i + C. (35)Hence, comparing (35) with (15) gives (34). ome non-elementary integrals and applications in probability theory R x α cos (cid:0) ηx β (cid:1) dx, R x α sin (cid:0) ηx β (cid:1) dx Proposition 2.9.
Let η and β be nonzero constants ( η = , β = ), α be someconstant different from − ( α = − ) and α = − β − . Then, Z x α cos (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) + βηx β sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i + C. (36)The proof is similar to the proof of Proposition 2.4, so it is omitted. Proposition 2.10.
Let η and β be nonzero constants ( η = , β = ), α be someconstant different from − ( α = − ) and α = − β − . Then Z x α sin (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) − βηx β cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i + C. (37)The proof of this proposition is also omitted since it is similar to that of Propo-sition 2.5. Theorem 2.11.
For any constants α, β and η , α + β + h cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) − βηx β sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i = (cid:20) e iηx β F (cid:16) α + β + β ; − iηx β (cid:17) + e − iηx β F (cid:16) α + β + β ; iηx β (cid:17)(cid:21) . (38)4 V. Nijimbere
Proof.
Euler’s identity cos (cid:0) ηx β (cid:1) = (cid:16) e iηx β + e − iηx β (cid:17) / Z x α cos (cid:16) ηx β (cid:17) dx = hZ x α e iηx β dx + Z x α e − iηx β dx i = x α + ( α + ) h e iηx β F (cid:16) α + β + β ; − iηx β (cid:17) + e − iηx β F (cid:16) α + β + β ; iηx β (cid:17)i + C. (39)Hence, Comparing (39) with (36) gives (38). Theorem 2.12.
For any constants α, β and η , α + β + h sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) + βηx β cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i = i (cid:20) e iηx β F (cid:16) α + β + β ; − iηx β (cid:17) − e − iηx β F (cid:16) α + β + β ; iηx β (cid:17)(cid:21) . (40) Proof.
Euler’s identity sin (cid:0) ηx β (cid:1) = (cid:16) e iηx β − e − iηx β (cid:17) / ( i ) and Proposition 2.2gives Z x α cos (cid:16) ηx β (cid:17) dx = hZ x α e iηx β dx + Z x α e − iηx β dx i = x α + i ( α + ) h e iηx β F (cid:16) α + β + β ; − iηx β (cid:17) − e − iηx β F (cid:16) α + β + β ; iηx β (cid:17)i + C. (41)Hence, Comparing (41) with (37) gives (40). Theorem 2.13.
For any constants α, β and η , e iηx β F (cid:16) α + β + β ; − iηx β (cid:17) ome non-elementary integrals and applications in probability theory = α + β + h cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) − βηx β sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i + iα + β + h sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) + βηx β cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i . (42) Proof.
Using the relation e iηx β = cos (cid:0) ηx β (cid:1) + i sin (cid:0) ηx β (cid:1) and Propositions 2.9and 2.10 yields Z x α e iηx β dx = Z x α cos (cid:16) ηx β (cid:17) dx + i Z x α sin (cid:16) ηx β (cid:17) dx = x α + ( α + )( α + β + ) h cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) + βηx β sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i + i x α + ( α + )( α + β + ) h sin (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17) − βηx β cos (cid:16) ηx β (cid:17) F (cid:16) α + β + β , α + β + β ; − η x β (cid:17)i + C. (43)Hence, comparing (43) with (15) (with η replaced by iη ) gives (42). In this section, Theorem 2.3 is used to generalize the gamma-type ( χ distribution,inverse gamma distribution) distribution and Gaussian-type distributions. Define a probability measure µ in terms of the Lebesgue measure dx as [3] dµ = µ ( dx ) = A g ( x ; α, η, β ) dx = f X ( x ; α, η, β ) dx, x ∈ [ , + ∞ ) , (44)6 V. Nijimbere where f X ( x ; α, η, β ) is the probability density function (p.d.f.) of some randomvariable X , g ( x ; α, η, β ) = x α e − ηx β , α = − , β = , α > − β − , (45)and A is a normalized constant which can be obtained using formula (19) in The-orem 2.3.After normalization, it is found that the p.d.f. of X is given by f X ( x ; α, η, β ) = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α e − ηx β , α = − , β = , α > − β − . (46)The distribution function of the random variable X can be obtained using Propo-sition 2.2 and is given by F X ( x ; α, η, β ) = µ { [ , x ) } = x Z f X ( u ; α, η, β ) du = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17) . (47)The n th moments ( M ( X n ) ) can, as well, be evaluated using formula (19) in The-orem 2.3 to obtain M ( X n ) = + ∞ Z x n f X ( x ; α, η, β ) dx = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) + ∞ Z x α + n e − ηx β dx = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) Γ (( α + β + n + ) /β )( α + n + ) η ( α + n + ) /β = Γ (( α + n + ) /β ) η α/β Γ (( α + ) /β ) . (48)These results are summarized in the following theorem. Theorem 3.1.
Let X be a random variable with the generalized gamma-type p.d.f. f X ( x ; α, η, β ) = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α e − ηx β , x ∈ R + , α = − , β = , α > − β − . (49) ome non-elementary integrals and applications in probability theory Then, the distribution function F X ( x ; α, η, β ) of the random variable X is givenby F X ( x ; α, η, β ) = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17) , (50) and the n th moments M ( X n ) of X are given by M ( X n ) = + ∞ Z x n f X ( x ; α, η, β ) dx = Γ (( α + n + ) /β ) η α/β Γ (( α + ) /β ) . (51)If Y is some gamma distribution random variable, then the random variable X = /Y is said to be an inverse gamma distribution random variable. Theinverse gamma distribution find applications in wireless communications, see forexample [4, 16]. Its distribution function may be evaluated using Theorem 3.1. Corollary 3.2.
Let X be a random variable with the inverse gamma distribution, X ∼ IG ( θ, η ) . Then, the distribution function F X ( x ; θ, η ) is given by F X ( x ; θ, η ) = − η θ Γ ( θ + ) x − θ e − η/x F (cid:16) θ + η/x (cid:17) , x > , θ > , η > , (52) while the n th moments M ( X n ) are given by M ( X n ) = η n Γ ( θ − n ) Γ ( θ + ) , n > θ. (53) Proof.
Setting α = − ( θ + ) , β = −
1, and using the fundamental theorem ofcalculus f X ( x ) = dF X dx = ddx R x f X ( u ) du and Proposition 2.2 gives the p.d.f. f X ( x ; α, η, β ) = η θ Γ ( θ ) x − ( θ + ) e − ηx − , x > , θ > , η > , (54)which is the p.d.f of the inverse gamma distribution. The n th moments M ( X n ) of X are obtained by setting α = − ( θ + ) and β = − V. Nijimbere
Consider a probability measure µ in terms of Lebesgue measure dx given by [3] dµ = µ ( dx ) = A g ( x ; α, η, β ) dx = f X ( x ; α, η, β ) dx, x ∈ R , (55)where, as before, f X ( x ; α, η, β ) is the p.d.f. of some random variable X , g ( x ; α, η, β ) = x α e − ηx β , α = − , β = , α > − β − , (56)is an even function of the variable x , and A is a normalized constant which can beobtained using formula (20) in Theorem 2.3.After normalization, the p.d.f. of f X is found to be f X ( x ; α, η, β ) = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α e − ηx β , x ∈ R , α = − , β = , α > − β − . (57)It is important to note that f X in this case is even, and so, a factor of 2 has toappear in the denominator. The distribution function F X can also be obtainedusing Proposition 2.2 and is thus given by F X ( x ; α, η, β ) = µ { ( −∞ , x ) } = x Z −∞ f X ( u ; α, η, β ) du = " − ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17) . (58)The moment ( M ( X n ) ) can be evaluated using formula (20) in Theorem 2.3 toobtain M ( X n ) = + ∞ Z −∞ x n f X ( x ; α, η, β ) dx = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) + ∞ Z −∞ x α + n e − ηx β dx = ( Γ (( α + n + ) /β ) η α/β Γ (( α + ) /β ) , if n is even . , if n is odd . (59)These results are summarized in the following theorem. ome non-elementary integrals and applications in probability theory Theorem 3.3.
Let X be a random variable with an even p.d.f. of the form f X ( x ; α, η, β ) = ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α e − ηx β , x ∈ R , α = − , β = , α > − β − . (60) Then, the distribution function F X ( x ; α, η, β ) of the random variable X is givenby F X ( x ; α, η, β ) = " − ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) x α + e − ηx β F (cid:16) α + β + β ; ηx β (cid:17) . (61) and the n th moments M ( X n ) of X are given by M ( X n ) = + ∞ Z −∞ x n f X ( x ; α, η, β ) dx = ( Γ (( α + n + ) /β ) η α/β Γ (( α + ) /β ) , if n is even . , if n is odd . (62)For example, setting α = , β = η = / f X ( x ) = ( / √ π ) e − x / ,and the mean of X is EX = M ( X ) = EX = M ( X ) =
1. So X ∼ N ( , ) distribution as expected.More general results can be achieved by introducing two additional parameters. Theorem 3.4.
Let X be a random variable with an even p.d.f.of the form f X ( x ; α, η, β, θ, σ ) = σ ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) × (cid:18) x − θσ (cid:19) α exp − η (cid:18) x − θσ (cid:19) β ! , x ∈ R , θ ∈ R , α = − , β = , α > − β − , σ > . (63) Then, the distribution function F X ( x ; α, η, β, θ, σ ) of the random variable X isgiven by F X ( x ; α, η, β, θ, σ ) = h − σ ( α + ) η ( α + ) /β Γ (( α + β + ) /β ) (cid:18) x − θσ (cid:19) α + exp − η (cid:18) x − θσ (cid:19) β ! F α + β + β ; η (cid:18) x − θσ (cid:19) β !i , (64)0 V. Nijimbere and the moments M ( X n ) of X are given by M ( X n ) = + ∞ Z −∞ x n f X ( x ; α, η, β ) dx = θ n Γ (( α + ) /β ) n X l = Γ (( α + l + ) /β ) C n l (cid:18) σθ η /β (cid:19) l , l ≤ n and ( l ) ∈ N , (65) where C n l = n ! / (( n − l ) ! ( l ) ! ) .Thus, the mean and the variance of X are respectively given by EX = M ( X ) = θ and var X = EX − ( EX ) = σ η /β Γ (( α + ) /β ) Γ (( α + ) /β ) . (66)Formula (65) is obtained by making the substitution u = ( x − θ ) /σ , and byapplying the binomial theorem and Theorem 2.3.A generalized Gaussian-type distribution may be derived by setting α = Corollary 3.5.
Let X be a random variable with the generalized Gaussian-typedistribution p.d.f. of the form f X ( x ; η, β, θ, σ ) = β σ η /β Γ ( /β ) exp − η (cid:18) x − θσ (cid:19) β ! , x ∈ R , θ ∈ R , β > , σ > , (67) and where β is even. Then, the distribution function F X ( x ; α, η, β, θ, σ ) of therandom variable X is given by F X ( x ; η, β, θ, σ ) = h − βσ η /β Γ ( /β ) × exp − η (cid:18) x − θσ (cid:19) β ! F β + β ; η (cid:18) x − θσ (cid:19) β !i , (68) and the n th moment M ( X n ) of X are given by M ( X n ) = + ∞ Z −∞ x n f X ( x ; η, β, θ, σ ) dx ome non-elementary integrals and applications in probability theory = θ n Γ ( /β ) n X l = Γ (( l + ) /β ) C n l (cid:18) σθ η /β (cid:19) l , l ≤ n and ( l ) ∈ N , (69) where, as before, C n l = n ! / (( n − l ) ! ( l ) ! ) . Thus, the mean and the variance of X are respectively given by EX = M ( X ) = θ and var X = EX − ( EX ) = σ η /β Γ ( /β ) Γ ( /β ) . (70) Moreover, if η ≥ / and β > , and since Γ ( /β ) < Γ ( /β ) , then the varianceof X (var X ) does satisfyvar X = EX − ( EX ) = σ η /β Γ ( /β ) Γ ( /β ) ≤ σ , (71) where σ is the variance of the Gaussian random variable. A formula for the n th moments of the Gaussian distribution can now be obtainedby setting β = , η = / α = Corollary 3.6.
Let X be a Gaussian random variable. Its n th moments M ( X n ) are thus given by the formula M ( X n ) = θ n √ π n X l = Γ ( l + / ) C n l (cid:18) σ θ (cid:19) l , l ≤ n and ( l ) ∈ N , (72) where θ ∈ R is the mean of the Gaussian random variable and σ > its variance,and as before, C n l = n ! / (( n − l ) ! ( l ) ! ) . Formulas for the integrals R x α e ηx β dx, R x α cosh (cid:0) ηx β (cid:1) dx, R x α sinh (cid:0) ηx β (cid:1) dx, R x α cos (cid:0) ηx β (cid:1) dx and R x α sin (cid:0) ηx β (cid:1) dx where α, η and β are real or complexconstants were obtained in terms of the confluent hypergeometric function F and the hypergeometric function F in section 2 (Propositions 2.2, 2.4, 2.5, 2.9and 2.10), and using hyperbolic and Euler identities, some identities involvingconfluent hypergeometric function F and hypergeometric function F werealso obtained in section 2 (Theorems 2.6-2.13). Having evaluated the integrals R + ∞−∞ x α e − ηx β dx, η > F , and formulas for the2 V. Nijimbere n th moments were obtained as well in section 3 (Theorems 3.1-3.4 and Corollaries3.5-3.6). The results obtained in this paper may, for example, may be used to con-struct better statistical tests than those already know (e.g. χ statistical tests andtests obtained based on the normal distribution). Bibliography [1] M. Abramowitz, I.A. Stegun,
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