Some Statistical Characteristics Depending on the Maximum Variance of Solution of Two Dimensional Stochastic Fredholm Integral Equation contains Two Gamma Processes Mohammad W. Muflih Noor A. A. Jabbar Dept. of Mathematics/College of Education For Pure Science(Ibn Al-Haitham) /Baghdad University Received in : 9 October 2013 , Accepted in 4December2013 Abstract In this paper, we find the two solutions of two dimensional stochastic Fredholm integral equations contain two gamma processes differ by the parameters in two cases and equal in the third are solved by the Adomain decomposition method. As a result of the solutions probability density functions and their variances at the time t are derived by depending upon the maximum variances of each probability density function with respect to the three cases. The auto covariance and the power spectral density functions are also derived. To indicate which of the three cases is the best, the auto correlation coefficients are calculated. Keywords: Two Dimensional Stochastic Fredholm Integral Equations, Gamma Process, Adomain Decomposition Method 368 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Introduction In this paper, the solution of the Stochastic Fredholm integral equation, for one and two dimensions is found analytically by a new method (beginning of 1980's) for solving linear and nonlinear integral (differential) equations for various kinds has been proposed by G.Adomian, the so called Adomian decomposition method, [1]. After that, in recent years many researchers interested as a final aim either by studying the existence and uniqueness of the solution of one or more dimensional integral equations (Balachandran et al., 2005; Milton et al., 1972) [4,7] or to find by using different methods of the modified quadrature the numerical solutions of this kind of equations on some definite closed interval to study a comparison between the numerical solutions and their exact solutions (Huda, 2007; AL-Sadany, 2008) [5]. While Vahidi and Mokhtari (2008) [9] use the Adomian decomposition method to compare this method with the classical successive method for solving system of linear Fredholm integral equations and Biazar and Rangbar (2007) [3] studied the comparison between Newton’s method and Adomian decomposition method for solving special Fredholm integral equations and Wahdan. M.M. (2011) [11] is interested not only in the solution of the supposing stochastic Fredholm integral equation but in concentrating in the derivation of many probability characteristics of the solution. In this paper, for combining the integral equations as an important branch of mathematics with some Statistical characteristics Stochastic Fredholm integral equations with gamma processes differ by the parameters in two cases and equal in the third. Our aim is not only interesting in the solution of the supposing stochastic Fredholm integral equation but we concentrate ourselves in the derivation of many statistical characteristics, of this solution (mean, variance, autocovariance function and power spectral density function) that is by depending on the maximum variances. 1. Preliminary The gamma process Г( w , α ,β , t ) is a lévy process whose marginal distribution at the time t > 0 is a gamma distribution with mean = αt β and variance = αt β2 Г(𝜔, 𝛼, 𝛽, 𝑡 ) = (𝛽)𝛼𝑡 Г(𝛼𝑡) 𝑤𝛼𝑡−1 𝑒−𝛽𝑤 , 𝑤 > 0, 𝑡 > 0 … (1) Where the parameter α controls the rate of jump arrivals (shape parameter) and the scaling parameter β inversely the jump size (scale parameter). [10] Moreover, the gamma process has the following properties: 1. Г(w, α, β, 0 ) = 0 . 2. For any 0≤ t0 ≤ t1 < … < tn < ∞ , n ≥ 1 Г(w, α, β, t1) − Г(w, α, β, t0), ⋯ , Г(w, α, β, tn) − Г(w, α, β, tn−1) are independent increments. 3. For any 0≤ s < t ; Г(w, α, β, t) − Г(w, α, β, s) have the same distribution as Г(w, α, β, t − s) . Now, we consider the following two dimensional systems of stochastic Fredholm integral Equation 2: 𝑌𝑖(𝑡, 𝑤) = Г𝑖(𝜔, 𝛼𝑖, 𝛽𝑖, 𝑡 ) + � �𝑘𝑖𝑗(𝑤, 𝑠, 𝑡)𝑌𝑗 2 𝑗=1 ∞ 0 (𝑠, 𝑡)𝑑𝑠 , 𝑖 = 1,2 … (2) where, - kij(w, s, t), i, j = 1, 2 are known stochastic kernels defined by t > 0, s S, S is a com and having respectively the supposing formulas Equation 3: 𝑘1𝑗(𝑡, 𝑠, 𝑤) = 𝑒− (𝑠+𝑤𝑡) , 𝑘2𝑗(𝑡, 𝑠, 𝑤) = 𝑒−�𝑠+𝑤𝑡 2� , 𝑗 = 1,2 … (3) - 𝑌𝑗(𝑠, 𝑡), 𝑗 = 1,2 are scalar functions defined for t>0, s>0. By substituting (1), (3) into (2), we get Equation 4: 369 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝑌1(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + � 𝑒−(𝑠+𝑤𝑡) (𝑌1(𝑡, 𝑠) + 𝑌2(𝑡, 𝑠))𝑑𝑠 ∞ 0 𝑌2(𝑡, 𝑤) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + � 𝑒−�𝑠+𝑤𝑡 2� (𝑌1(𝑡, 𝑠) + 𝑌2(𝑡, 𝑠))𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ … (4) Remark (1) In this study, three cases for the parameters (𝛼, 𝛽) should be considered : (1) 𝛼1 > 𝛽1 , 𝛼1 = 1 , 𝛽1 = 0.5 , 𝛼2 > 𝛽2 , 𝛼2 = 2 , 𝛽2 = 1.5 (2)𝛼1 = 𝛽1 = 0.5 , 𝛼2 > 𝛽2 , 𝛼2 = 1.5, 𝛽2 = 1 (3)𝛼1 = 𝛽1 = 𝛼2 = 𝛽2 = 1 � Now, to find the stochastic solutions of the system (4), Adomian decomposition method should be used which briefly depends on the following steps Equation 5-7,[9] 𝑌10(𝑡, 𝑤) = Г1(𝜔, 𝛼1, 𝛽1, 𝑡 ) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 𝑌20(𝑡, 𝑤) = Г2(𝜔, 𝛼2, 𝛽2, 𝑡 ) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 ⎭ ⎪ ⎬ ⎪ ⎫ … (5) 𝑌10(𝑡, 𝑠) = Г1(𝑠, 𝛼1, 𝛽1, 𝑡 ) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑠𝛼1𝑡−1 𝑒−𝛽1𝑠 𝑌20(𝑡, 𝑠) = Г2(𝑠, 𝛼2, 𝛽2, 𝑡 ) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑠𝛼2𝑡−1 𝑒−𝛽2𝑠 ⎭ ⎪ ⎬ ⎪ ⎫ … (6) And, 𝑌1,𝑚+1(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡) (𝑌1𝑚(𝑡, 𝑠) + 𝑌2𝑚(𝑡, 𝑠)) 𝑑𝑠 ∞ 0 𝑌2,𝑚+1(𝑡, 𝑤) = � 𝑒−�𝑠+𝑤𝑡 2� (𝑌1𝑚(𝑡, 𝑠) + 𝑌2𝑚(𝑡, 𝑠)) 𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ , 𝑚 = 0,1,2, … … (7) That is to get the following two stochastic solutions Equation (8): 𝑌1(𝑡, 𝑤) = 𝑌10(𝑡, 𝑤) + � 𝑌1𝑛(𝑡, 𝑤) ∞ 𝑛=1 𝑌2(𝑡, 𝑤) = 𝑌20(𝑡, 𝑤) + � 𝑌2𝑛(𝑡, 𝑤) ∞ 𝑛=1 ⎭ ⎪ ⎬ ⎪ ⎫ … (8) So, for m = 0, by (6) and (7): 370 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝑌11(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡)(𝑌10(𝑡, 𝑠) + 𝑌20(𝑡, 𝑠) )𝑑𝑠 ∞ 0 𝑌21(𝑡, 𝑤) = � 𝑒−�𝑠+𝑤𝑡 2�� 𝑌10(𝑡, 𝑠) + 𝑌20(𝑡, 𝑠)�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌11(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡) � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑠𝛼1𝑡−1 𝑒−𝛽1𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑠𝛼2𝑡−1 𝑒−𝛽2𝑠�𝑑𝑠 ∞ 0 𝑌21(𝑡, 𝑤) = � 𝑒−�𝑠+𝑤𝑡 2� � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑠𝛼1𝑡−1 𝑒−𝛽1𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑠𝛼2𝑡−1 𝑒−𝛽2𝑠�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌11(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑒−𝑤𝑡 � 𝑠𝛼1𝑡−1𝑒−(𝛽1+1)𝑠𝑑𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑒−𝑤𝑡 � 𝑠𝛼2𝑡−1𝑒−(𝛽2+1)𝑠𝑑𝑠 ∞ 0 ∞ 0 𝑌21(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑒−𝑤𝑡 2 � 𝑠𝛼1𝑡−1𝑒−(𝛽1+1)𝑠𝑑𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑒−𝑤𝑡 2 � 𝑠𝛼2𝑡−1𝑒−(𝛽2+1)𝑠𝑑𝑠 ∞ 0 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌11(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑒−𝑤𝑡 � 𝑠𝛼1𝑡−1𝑒 − 𝑠 1 𝛽1+1⁄ 𝑑𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑒−𝑤𝑡 � 𝑠𝛼2𝑡−1𝑒 − 𝑠 1 𝛽2+1⁄ 𝑑𝑠 ∞ 0 ∞ 0 𝑌21(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑒−𝑤𝑡 2 � 𝑠𝛼1𝑡−1𝑒 − 𝑠 1 𝛽1+1⁄ 𝑑𝑠 + (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑒−𝑤𝑡 2 � 𝑠𝛼2𝑡−1𝑒 − 𝑠 1 𝛽2+1⁄ 𝑑𝑠 ∞ 0 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ or 𝑌11(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 𝑌21(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 2 ⎭ ⎪ ⎬ ⎪ ⎫ … (9) and for m = 1 , by (7) and (9): 𝑌12(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡)(𝑌11(𝑡, 𝑠) + 𝑌21(𝑡, 𝑠) )𝑑𝑠 ∞ 0 𝑌22(𝑡, 𝑤) = � 𝑒−�𝑠+𝑤𝑡 2�� 𝑌11(𝑡, 𝑠) + 𝑌21(𝑡, 𝑠)�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌12(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡) ��� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑠𝑡 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑠𝑡 2 �𝑑𝑠 ∞ 0 𝑌22(𝑡, 𝑤) = � 𝑒−�𝑠+𝑤𝑡 2� ��� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑠𝑡 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑠𝑡 2�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌12(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �𝑒−𝑤𝑡 �� 𝑒−(1+𝑡)𝑠 + 𝑒−�1+𝑡 2�𝑠 �𝑑𝑠 ∞ 0 𝑌22(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 2 �� 𝑒−(1+𝑡)𝑠 + 𝑒−�1+𝑡 2�𝑠�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 371 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 or 𝑌12(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑤𝑡 𝑌22(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑤𝑡 2 ⎭ ⎪ ⎬ ⎪ ⎫ … (10) and for m = 2 , by (7) and (10): 𝑌13(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡)�𝑌12(𝑡, 𝑠) + 𝑌22(𝑡, 𝑠)�𝑑𝑠 ∞ 0 𝑌23(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡2)� 𝑌12(𝑡, 𝑠) + 𝑌22(𝑡, 𝑠)�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ 𝑌13(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡) ⎝ ⎜ ⎛ �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑠𝑡 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑠𝑡 2 ⎠ ⎟ ⎞ 𝑑𝑠 ∞ 0 𝑌23(𝑡, 𝑤) = � 𝑒− (𝑠+𝑤𝑡2) ⎝ ⎜ ⎛ �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑠𝑡 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑠𝑡 2 ⎠ ⎟ ⎞ 𝑑𝑠 ∞ 0 ⎭ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎫ 𝑌13(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑒−𝑤𝑡 ��𝑒−(1+𝑡)𝑠 + 𝑒−�1+𝑡 2�𝑠�𝑑𝑠 ∞ 0 𝑌23(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) �𝑒−𝑤𝑡 2 ��𝑒−(1+𝑡)𝑠 + 𝑒−�1+𝑡 2�𝑠�𝑑𝑠 ∞ 0 ⎭ ⎪ ⎬ ⎪ ⎫ or 𝑌13(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 2 𝑒−𝑤𝑡 𝑌23(𝑡, 𝑤) = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 2 𝑒−𝑤𝑡 2 ⎭ ⎪ ⎬ ⎪ ⎫ … (11) and by repeating iterations for m=3,4,… and adding (9) ,(10) and(11) we get : �𝑌1𝑛(𝑡, 𝑤) ∞ 𝑛=1 = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 � � 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑘∞ 𝑘=0 � 𝑌2𝑛(𝑡, 𝑤) = ∞ 𝑛=1 �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 2 �� 𝑡2 + 𝑡 + 2 (1 + 𝑡)(1 + 𝑡2) � 𝑘∞ 𝑘=0 ⎭ ⎪ ⎬ ⎪ ⎫ Where ∑ � 𝑡 2+𝑡+2 (1+𝑡)(1+𝑡2) � k ∞ k=0 is a geometric series converges for t> 1, hence Equation (12) : � 𝑌1𝑛(𝑡, 𝑤) ∞ 𝑛=1 = �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � �𝑌2𝑛(𝑡, 𝑤) = ∞ 𝑛=1 �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 2 � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � ⎭ ⎪ ⎬ ⎪ ⎫ … (12) Finally, by substituting (5) and (12) into (8) which represents the two stochastic solutions of (4), we get Equation (13): 372 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝑌1(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � 𝑌2(𝑡, 𝑤) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � 𝑒−𝑤𝑡 2 � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � ⎭ ⎪ ⎬ ⎪ ⎫ … (13) where, w > 0 , t  , αi , βi > 0 , i = 1,2 Moreover, the stochastic solutions (13) can be considered as a stochastic solutions over the interval 01. 2. Statistical Characteristics of the Stochastic Solutions In order to derive the statistical characteristics of the two stochastic solutions (13) over the interval 01, it must be that each of them is a probability density function (p.d.f.) of (t,w). So, we multiply them respectively by A and B and equate their integrals by one that is to find A and B which make each stochastic solution is a p.d.f., i.e., we write: �𝐴 𝑌1(𝑡, 𝑤) 𝑑𝑤 = 1 1 0 , �𝐵 𝑌2(𝑡, 𝑤) 𝑑𝑤 = 1 1 0 �𝐴 � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � 𝑒−𝑤𝑡 �𝑑𝑤 = 1 1 0 �𝐵� (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � 𝑒− 𝑤 𝑡2 �𝑑𝑤 = 1 1 0 𝐴 (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) �𝑤𝛼1𝑡−1 1 0 �1 − 𝛽1𝑤 1! + (𝛽1𝑤)2 2! − (𝛽1𝑤)3 3! + ⋯�𝑑𝑤 + 𝐴�� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � � 𝑒− 𝑤 𝑡 1 0 𝑑𝑤 = 1 𝐵 (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) �𝑤𝛼2𝑡−1 1 0 �1 − 𝛽2𝑤 1! + (𝛽2𝑤)2 2! − (𝛽2𝑤)3 3! + ⋯�𝑑𝑤 + 𝐵 �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � �𝑒−𝑊𝑡 2 1 0 𝑑𝑤 = 1 𝐴 (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) � � (−𝛽1)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼1𝑡+𝑛−1𝑑𝑤 + 𝐴�� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � � 𝑒−𝑤𝑡 1 0 𝑑𝑤 = 1 𝐵 (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) � � (−𝛽2)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼2𝑡+𝑛−1𝑑𝑤 + 𝐵 �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � �𝑒−𝑤𝑡 2 1 0 𝑑𝑤 = 1 𝐴 (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) � (−𝛽1)𝑛 𝑛! (𝛼1𝑡 + 𝑛) ∞ 𝑛=0 + 𝐴�� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡(𝑡3 − 1) � (1 − 𝑒−𝑡) = 1 𝐵 (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) � (−𝛽2)𝑛 𝑛! (𝛼2𝑡 + 𝑛) ∞ 𝑛=0 + 𝐵 �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2 𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡2(𝑡3 − 1) � �1 − 𝑒−𝑡 2� = 1 Hence 𝐴 = 1 (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡)∞𝑛=0 373 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝐵 = 1 (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 and let 𝜁1(𝑡, 𝑤) = 𝐴 𝑌1(𝑡, 𝑤) 𝜁2(𝑡, 𝑤) = 𝐵 𝑌2(𝑡, 𝑤) Hence, the two stochastic solutions (13) are probability density functions of (t,w) that is when: 𝜁1(𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) (𝑡3 − 1) � 𝑒 −𝑤𝑡 (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡)∞𝑛=0 …(14.a) 𝜁2(𝑡, 𝑤) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) (𝑡3 − 1) � 𝑒 −𝑤𝑡2 (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 …(14.b) For both Equation (14.a) and (14.b) 0 < w ≤1,t  , αi > 0, βi > 0, i = 1,2. or 𝜁1(𝑡, 𝑤) = 𝜃1(𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + 𝛿1(𝑡)𝑒−𝑤𝑡 𝜁2(𝑡, 𝑤) = 𝜃2(𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + 𝛿2(𝑡)𝑒−𝑤𝑡 2 where 𝜃1(𝑡) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡)∞𝑛=0 𝛿1(𝑡) = �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡)∞𝑛=0 𝜃2(𝑡) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 𝛿2(𝑡) = �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 374 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 3. First and Second Means of 𝜻𝟏(𝒕, 𝒘) and 𝜻𝟐(𝒕, 𝒘) First Mean: 𝐸𝜁1 (𝑡, 𝑤) = �𝑤[ 𝜁1(𝑡, 𝑤)]𝑑𝑤 1 0 𝐸𝜁2 (𝑡, 𝑤) = �𝑤[ 𝜁2(𝑡, 𝑤)]𝑑𝑤 1 0 ⎭ ⎪ ⎬ ⎪ ⎫ we start by the first mean of ζ1(t, w): 𝐸𝜁1 (𝑡, 𝑤) = �𝑤 1 0 [𝜃1(𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + 𝛿1(𝑡)𝑒−𝑤𝑡] 𝑑𝑤 = 𝜃1(𝑡) �𝑤𝛼1𝑡 1 0 �1 − 𝛽1𝑤 1! + (𝛽1𝑤)2 2! − (𝛽1𝑤)3 3! + ⋯�𝑑𝑤 + 𝛿1(𝑡) �𝑤 𝑒−𝑤𝑡𝑑𝑤 1 0 = 𝜃1(𝑡) � � (−𝛽1)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼1𝑡+𝑛𝑑𝑤 + 𝛿1(𝑡) �𝑤 𝑒−𝑤𝑡𝑑𝑤 1 0 = 𝜃1(𝑡) � (−𝛽1)𝑛 𝑛! (𝛼1𝑡 + 𝑛 + 1) ∞ 𝑛=0 + 𝛿1(𝑡) � 1 − (1 + 𝑡) 𝑒−𝑡 𝑡2 � or 𝐸𝜁1 (𝑡, 𝑤) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1)𝑛 𝑛! (𝛼1𝑡 + 𝑛 + 1) ∞ 𝑛=0 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 �� 1 − (1 + 𝑡) 𝑒−𝑡 𝑡2 � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡)∞𝑛=0 …(15) while, the first mean of ζ2(t, w): 𝐸𝜁2 (𝑡, 𝑤) = �𝑤 1 0 [𝜃2(𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + 𝛿2(𝑡)𝑒−𝑤𝑡 2 ] 𝑑𝑤 = 𝜃2(𝑡) �𝑤𝛼2𝑡 1 0 �1 − 𝛽2𝑤 1! + (𝛽2𝑤)2 2! − (𝛽2𝑤)3 3! + ⋯�𝑑𝑤 + 𝛿2(𝑡) �𝑤 𝑒−𝑤𝑡 2 𝑑𝑤 1 0 = 𝜃2(𝑡) � � (−𝛽2)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼2𝑡+𝑛𝑑𝑤 + 𝛿2(𝑡) �𝑤 𝑒−𝑤𝑡 2 𝑑𝑤 1 0 = 𝜃2(𝑡) � (−𝛽2)𝑛 𝑛! (𝛼2𝑡 + 𝑛 + 1) ∞ 𝑛=0 + 𝛿2(𝑡) � 1 − (𝑡2 + 1)𝑒−𝑡 2 𝑡4 � or 375 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝐸𝜁2 (𝑡, 𝑤) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2)𝑛 𝑛! (𝛼2𝑡 + 𝑛 + 1) ∞ 𝑛=0 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � � 1 − (𝑡2 + 1)𝑒−𝑡 2 𝑡4 � (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 (16)... Second Mean: 𝐸𝜁1 (𝑡, 𝑤 2) = ∫ 𝑤2[ 𝜁1(𝑡, 𝑤)]𝑑𝑤 1 0 𝐸𝜁2 (𝑡, 𝑤 2) = ∫ 𝑤2[ 𝜁2(𝑡, 𝑤)]𝑑𝑤 1 0 � we start by the first mean of ζ1(t, w): 𝐸𝜁1 (𝑡, 𝑤 2) = �𝑤2 1 0 [𝜃1(𝑡) 𝑤𝛼1𝑡−1 𝑒−𝛽1𝑤 + 𝛿1(𝑡)𝑒−𝑤𝑡] 𝑑𝑤 = 𝜃1(𝑡) �𝑤𝛼1𝑡+1 1 0 �1 − 𝛽1𝑤 1! + (𝛽1𝑤)2 2! − (𝛽1𝑤)3 3! + ⋯�𝑑𝑤 + 𝛿1(𝑡) �𝑤2 𝑒−𝑤𝑡𝑑𝑤 1 0 = 𝜃1(𝑡) � � (−𝛽1)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼1𝑡+𝑛+1𝑑𝑤 + 𝛿1(𝑡) �𝑤2 𝑒−𝑤𝑡𝑑𝑤 1 0 = 𝜃1(𝑡) � (−𝛽1)𝑛 𝑛! (𝛼1𝑡 + 𝑛 + 2) ∞ 𝑛=0 + 𝛿1(𝑡) � 2 − (𝑡3 + 2𝑡2 + 2) 𝑒−𝑡 𝑡3 � or 𝐸𝜁1 (𝑡, 𝑤 2) = (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1)𝑛 𝑛! (𝛼1𝑡 + 𝑛 + 2) ∞ 𝑛=0 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 �� (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 �� 2 − (𝑡3 + 2𝑡2 + 2) 𝑒−𝑡 𝑡3 � (𝛽1)𝛼1𝑡 Г(𝛼1𝑡) ∑ (−𝛽1) 𝑛 𝑛! (𝛼1𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡(𝑡3 − 1) � (1 − 𝑒 −𝑡) ∞𝑛=0 …(17) while, the first mean of ζ2(t, w): 𝐸𝜁2 (𝑡, 𝑤 2) = �𝑤2 1 0 [𝜃2(𝑡) 𝑤𝛼2𝑡−1 𝑒−𝛽2𝑤 + 𝛿2(𝑡)𝑒−𝑤𝑡 2 ] 𝑑𝑤 = 𝜃2(𝑡) �𝑤𝛼2𝑡+1 1 0 �1 − 𝛽2𝑤 1! + (𝛽2𝑤)2 2! − (𝛽2𝑤)3 3! + ⋯�𝑑𝑤 + 𝛿2(𝑡) �𝑤2 𝑒−𝑤𝑡 2 𝑑𝑤 1 0 = 𝜃2(𝑡) � � (−𝛽2)𝑛 𝑛! ∞ 𝑛=0 1 0 𝑤𝛼2𝑡+𝑛+1𝑑𝑤 + 𝛿2(𝑡) �𝑤2 𝑒−𝑤𝑡 2 𝑑𝑤 1 0 = 𝜃2(𝑡) � (−𝛽2)𝑛 𝑛! (𝛼2𝑡 + 𝑛 + 2) ∞ 𝑛=0 + 𝛿2(𝑡) � 2 − (𝑡4 + 2𝑡2 + 2)𝑒−𝑡 2 𝑡6 � or 𝐸𝜁2 (𝑡, 𝑤 2) = (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2)𝑛 𝑛! (𝛼2𝑡 + 𝑛 + 2) ∞ 𝑛=0 + �� 𝛽1 𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 � � (1 + 𝑡)(1 + 𝑡2) 𝑡3 − 1 � � 2 − (𝑡4 + 2𝑡2 + 2)𝑒−𝑡 2 𝑡6 � (𝛽2)𝛼2𝑡 Г(𝛼2𝑡) ∑ (−𝛽2) 𝑛 𝑛! (𝛼2𝑡 + 𝑛) + �� 𝛽1𝛽1 + 1 � 𝛼1𝑡 + � 𝛽2𝛽2 + 1 � 𝛼2𝑡 ��(1 + 𝑡)(1 + 𝑡 2) 𝑡2(𝑡3 − 1) � �1 − 𝑒 −𝑡2�∞𝑛=0 …(18) 376 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Calculations of the variances of the probability density functions (p.d.f’s) (14.a), (14.b)by the formula Varζ(t, w) = Eζ(t, w2) − �Eζ(t, w)� 2 for the three cases (Remark 1) when t>1in table 1 and 2 . It is noted that, these variances in the three cases are slowly decreasing . Table (1) Variances of 𝛇𝟏(𝐭, 𝐰) α1 =β1 = α2= β2=1 α1 =β1 =0.5, α2> β2, α2=1.5, β2=1 α1 >β1 , α1=1, β1=0.5 α2> β2 , α2=2, β2=1.5 t 0.0785977 0.0794946 0.0789515 1.1 0.0788867 0.0777177 0.0804149 1.6 0.0836413 0.0791683 0.0848566 2.1 0.0898906 0.081976 0.0892143 2.6 0.0962628 0.0848037 0.0909379 3.1 0.1011318 0.0875569 0.0877991 3.6 0.1023229 0.0905387 0.079171 4.1 0.0979512 0.094079 0.0667281 4.6 0.0876862 0.0984085 0.0534102 5.1 0.0733108 0.1036144 0.0416486 5.6 0.057837 0.1096082 0.0324904 6.1 0.0439211 0.1160917 0.0258588 6.6 0.0329128 0.1225278 0.0211867 7.1 0.0249311 0.1281411 0.0178657 7.6 0.0194327 0.1319801 0.0154254 8.1 0.0157145 0.1330655 0.0135534 8.6 0.0131734 0.1306163 0.012057 9.1 . . . 0.0000307 0.0000307 0.0000307 180.6 . . . 0 0 0 500 377 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Table 3.2.1 Variances of 𝛇𝟏(𝐭, 𝐰) α1 =β1 = α2= β2=1 α1 =β1 =0.5, α2> β2, α2=1.5, β2=1 α1 >β1 , α1=1, β1=0.5 α2> β2 , α2=2, β2=1.5 t 0.0777596 0.0782779 0.0789953 1.1 0.0779569 0.0800432 0.0885412 1.6 0.0910475 0.0878936 0.1049919 2.1 0.1048118 0.0901711 0.1145189 2.6 0.1160848 0.078143 0.103725 3.1 0.1270881 0.0544222 0.0720713 3.6 0.1394045 0.0305826 0.0379876 4.1 0.1524344 0.0145272 0.0161455 4.6 0.163068 0.0062519 0.0061193 5.1 0.1661121 0.0026571 0.0023558 5.6 0.1565736 0.0012327 0.0010656 6.1 0.1335063 0.000673 0.0006066 6.6 0.1019399 0.0004323 0.0004104 7.1 . . . 0 0 0 100 The maximum variances from table 1 and 2 are presented in the following tables Table (3) Presents the maximum variances from both Table 1and 2 α1 =β1 = α2= β2=1 α1 =β1 =0.5, α2> β2, α2=1.5, β2=1 α1 >β1 , α1=1, β1=0.5, α2> β2 , α2=2, β2=1.5 t p.d.f ... ... 0.1023229 ... 0.1330655 … 0.0909379 … … 3.1 8.6 4.1 𝜁1(𝑡, 𝑤) ... 0.1661121 0.0901711 … 0.1145189 … 2.6 5.6 𝜁2 (𝑡, 𝑤) Furthermore, the probability density functions (14a) and (14b) with respect to the three cases of Remark 1) can be rewritten respectively as follows :( 1. when 𝜶𝟏 > 𝜷𝟏 ∶ 𝜶𝟏 = 𝟏 , 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 ∶ 𝜶𝟐 = 𝟐 , 𝜷𝟐 = 𝟏. 𝟓 𝜁1(3.1, 𝑤) = 1.1331142 𝑤2.1 𝑒−0.5𝑤 + 2.4294245 𝑒−3.1 𝑤 … (19. a) 𝜁2(2.6, 𝑤) = 5.5033493 𝑤4.2 𝑒−1.5𝑤 + 4.6848782 𝑒−6.76 𝑤 … (19. b) where t =3.1,2.6 ,0 < w < 1. 2. when 𝛂𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏 𝜁1(8.6, 𝑤) = 2.7582866 𝑤3.3 𝑒−0.5𝑤 + 4.9123986 𝑒−8.6 𝑤 … (19. c) 𝜁2(2.6, 𝑤) = 1.9151574 𝑤2.9 𝑒−𝑤 + 5.2464038𝑒−6.76 𝑤 … (19. d) . where t = 8.6,2.6 ,0 < w <1 378 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 3. when 𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 𝜁1(4.1, 𝑤) = 2.7357416 𝑤3.1 𝑒−𝑤 + 2.9068758 𝑒−4.1 𝑤 … (19. e) 𝜁2(5.6, 𝑤) = 5.6779497 𝑤4.6 𝑒−𝑤 + 17.626968 𝑒−31.36 𝑤 … (19. f) where t = 4.1,5.6 ,0 < w <1 Figures (1) - (3) represent the graphs of two pairs of p.d.f’s ζ1(t, w), ζ2(t, w). Figure (1) The Curve of 𝜻𝟏(3.1, w), 𝜻𝟐(2.6, w) when 𝜶𝟏 > 𝜷𝟏 ∶ 𝜶𝟏 = 𝟏 , 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 ∶ 𝜶𝟐 = 𝟐 , 𝜷𝟐 = 𝟏. 𝟓 Figure (2) The Curve of 𝜻𝟏(8.6, w), 𝜻𝟐(2.6, w) when 𝛂𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏 Figure (3) The Curve of 𝜻𝟏(4.1, w), 𝜻𝟐(5.6, w) when 𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 1.4 Autocovariance Function 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 w ζ 1( 3.1 ,w) , ζ 2 (2. 6,w ) ζ1(3.1,w) ζ2(2.6,w) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 w ζ 1(8 .6,w ), ζ 2 (2.6 ,w) ζ1(8.6,w) ζ2(2.6,w) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 w ζ 1( 4.1 ,w) , ζ 2 (5. 6,w ) ζ1(4.1,w) ζ2(5.6,w) 379 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 For any s>t , s – t = τ , the autocovariance function h(τ) of the Independent p.d.f functions 𝜁(t, w), 𝜁 (t+τ,w ) is an even function deepens on the difference |τ| = |s – t| = |t – s| and can be found as follows :[2] ℎ(𝜏) = 𝐸ζ[(𝑡, 𝑤)(𝑡 + 𝜏, 𝑤)] – 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤) = 𝐸ζ(𝑡, 𝑤2) + 𝐸ζ[(𝑡, 𝑤)(𝑡 + 𝜏, 𝑤)]– 𝐸ζ(𝑡, 𝑤2)– 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤) = 𝐸ζ(𝑡, 𝑤2) + 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤)– 𝐸ζ(𝑡, 𝑤2)– 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤) = 𝐸ζ (𝑡, 𝑤2) + 𝐸ζ(𝑡, 𝑤)�𝐸ζ(𝑡 + 𝜏, 𝑤)– 𝐸ζ(𝑡, 𝑤)�– 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤) = 𝐸ζ(𝑡, 𝑤2) + 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤)– {𝐸ζ(𝑡, 𝑤)}2 – 𝐸ζ(𝑡, 𝑤)𝐸ζ(𝑡 + 𝜏, 𝑤) or, ℎ(𝜏) = 𝐸ζ(𝑡, 𝑤2)– {𝐸ζ(𝑡, 𝑤)}2 = varζ(t, w), 𝜏 > 0 …(20) So, by Table 3 and Equation (20) either when ζ1(t, w), t = 3.1,8.6,4.1 or ζ2(t, w),t = 2.6,5.6 the autocovariance functions for them with respect to the three cases (Remark 1) are presented in Table 4. Table (4) Presents the Autocovariance Functions of 𝛇(𝐭, 𝐰), and 𝛇(𝐭 + 𝛕, 𝐰), α1 =β1 = α2= β2=1 α1 =β1 =0.5, α2> β2, α2=1.5, β2=1 α1 >β1 , α1=1, β1=0.5, α2> β2 , α2=2, β2=1.5 t p.d.f ... ... 0.1023229 ... 0.1330655 … 0.0909379 … … 3.1 8.6 4.1 𝜁1(𝑡, 𝑤) ... 0.1661121 0.0901711 … 0.1145189 … 2.6 5.6 𝜁2(𝑡, 𝑤) 1.5 Power Spectral Density Function let{X(t) , t ϵ T} be a stationary process with autocovariance function h(τ). The power spectral density function f𝜁(λ) ( p.s.d.f) of any probability density function 𝜁(t, w) is an even function represents the average power in this p.d.f at the angular frequency 0 < λ ≤ 2nπ , n ϵ I+and can be found by Khinchin’s formula as follows[6]: 𝑓𝜁(𝜆) = 1 2𝜋 � ℎ(𝜏)𝑒−𝑖𝜆𝜏 ∞ −∞ 𝑑𝜏 … (21) = ℎ(𝜏) 2𝜋 �(cos 𝜆𝜏 ∞ −∞ − 𝑖 sin 𝜆𝜏) 𝑑𝜏 and for τ = s – t > 0. = ℎ(𝜏) 𝜋 � (cos 𝜆𝜏 − 𝑖 sin 𝜆𝜏) 𝑑𝜏 s – t 0 or 𝑓𝜁(𝜆) = ℎ(𝜏) 𝜋 sin[𝜆(𝑠 − 𝑡)] 𝜆 , 0 < 𝜆 ≤ 2𝑛𝜋 , 𝑛 𝜖 𝐼+ … (22) 1. 𝜶𝟏 > 𝜷𝟏 , 𝜶𝟏 = 𝟏 , 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟐 , 𝜷𝟐 = 𝟏. 𝟓 𝑓𝜁1 (𝜆) = 0.0909379 𝜋 sin[𝜆(𝑠 − 3.1)] 𝜆 , 0 < 𝜆 < 2𝜋 … (22. a) 380 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 𝑓𝜁2 (𝜆) = 0.1145189 𝜋 sin[𝜆(𝑠 − 2.6)] 𝜆 ,0 < 𝜆 < 2𝜋 … (22. b) 2. 𝜶𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏 𝑓𝜁1 (𝜆) = 0.1330655 𝜋 sin[𝜆(𝑠 − 8.6)] 𝜆 , 0 < 𝜆 < 2𝜋 … (22. c) 𝑓𝜁2 (𝜆) = 0.0901711 𝜋 sin[𝜆(𝑠 − 2.6)] 𝜆 , 0 < 𝜆 < 2𝜋 … (22. d) 𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 𝑓𝜁1 (𝜆) = 0.1023229 𝜋 sin[𝜆(𝑠 − 4.1)] 𝜆 ,0 < 𝜆 < 2𝜋 … (22. e) 𝑓𝜁2 (𝜆) = 0.1661121 𝜋 sin[𝜆(𝑠 − 5.6)] 𝜆 ,0 < 𝜆 < 2𝜋 … (22. f) Figures (4) – (6) represent respectively the graphs of two pairs (fζ1(λ), fζ2(λ)) corresponding the three cases (Remark1) and s =(9.1,5.6) ,(9.1,5.6) and (6,10)respectively just be chosen to complete the figures . Figure (4) The Curve of 𝒇𝜻𝟏(λ), 𝒇𝜻𝟐(λ) for 0 < λ ≤ 2π, when s = (9.1,5.6) , 𝜶𝟏 > 𝜷𝟏 ∶ 𝜶𝟏 = 𝟏 , 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 ∶ 𝜶𝟐 = 𝟐 , 𝜷𝟐 = 𝟏. 𝟓 Figure (5) The Curve of 𝒇𝜻𝟏(λ), 𝒇𝜻𝟐(λ) for 0 < λ ≤ 2π, when s = (9.1,5.6) , 𝛂𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏 0 50 100 150 200 250 300 350 400 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x 10 -3 λ f ζ1( λ), f ζ2( λ) fζ1(λ),s=9.1 fζ2(λ),s=5.6 0 50 100 150 200 250 300 350 400 -4 -2 0 2 4 6 8 10 12 14 16 x 10 -4 λ f ζ1( λ), f ζ2( λ) fζ1(λ),s=9.1 fζ2(λ),s=5.6 381 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Figure (6) The Curve of 𝒇𝜻𝟏 (λ), 𝒇𝜻𝟐 (λ) for 0 < λ ≤ 2π, when s = (6,10) , 𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 1.6 Correlation Coefficients By the known Pearson's correlation coefficient formula for dependent two random variables X, Y:[8] ρx,y = E(XY) − E(X)E(Y) �var(X)var(Y) and since each pair of any two p.d.f’s ζ1(t, w ),ζ2(t, w) (22.a) to 22.f) with respect to the three cases( Remark 1) are also dependent. By writing: E(XY) = Expectation of the product of ζ1(t, w ) by ζ2(t, w) E(X) E(Y) = Product of expectation of ζ1(t, w ) by ζ2(t, w) Var(X) = Maximum variance of ζ1(t, w ) Var(Y) = Maximum variance of ζ2(t, w) Where, 1. 𝛂𝟏 > 𝛃𝟏: 𝛂𝟏 = 𝟏 , 𝛃𝟏 = 𝟎. 𝟓 , 𝛂𝟐 > 𝛃𝟐: 𝛂𝟐 = 𝟐 , 𝛃𝟐 = 𝟏. 𝟓 𝜌𝜁1𝜁2 = 0.179068 − (0.3916165)(0.3499049) �(0.0909379)(0.1145189) = 0.4119515 2. 𝜶𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏 𝜌𝜁1𝜁2 = 0.173058 − (0.4088372)(0.2859645) �(0.1330655)(0.09017106) = 0.5125611 3. 𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 𝜌𝜁1𝜁2 = 0.2672499 − (0.3931854)(0.38145) �(0.1023229)(0.1661121) = 0.8994918 0 50 100 150 200 250 300 350 400 -1 0 1 2 3 4 5 x 10 -3 λ f ζ1 ( λ) ,f ζ( λ) fζ1(λ),s=6 fζ2(λ),s=10 382 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Conclusion With respect to the considering three cases (Remark 1), the maximum variances of the resulting probability density function ζ1(t, w) (14a) is the greatest with respect to the in the second case (𝜶𝟏 = 𝜷𝟏 = 𝟎. 𝟓 , 𝜶𝟐 > 𝜷𝟐 , 𝜶𝟐 = 𝟏. 𝟓, 𝜷𝟐 = 𝟏) when (t = 8.6) while the probability density function ζ2(t, w) (14b) is the greatest with respect to the third case (𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 ) when (t = 5.6) Furthermore, the correlation coefficient between any pair of the probability density functions ζ1(t, w) , ζ2(t, w) is the greatest with respect to the third case (𝜶𝟏 = 𝜷𝟏 = 𝜶𝟐 = 𝜷𝟐 = 𝟏 ). As a recommendation, it is possible to use the numerically or analytically solutions for any kind of integral equations to study some statistical properties of those solutions that is firstly by deriving the p.d.f’s. For that, we suggest to consider other cases of the values of the parameters of the gamma processes that differ by the cases which are studied in this article. References 1. Adomian, G., (1994). "Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Dordrecht, Holland. 2. Basu, A.K.,(2003). “Introduction to Stochastic Process”, Alph International Ltd, Pangbourne, England. 3. Biazar, J. and Rangbar, A. (2007)."A comparison between newton’s method and A.D.M for solving special Fredholm integral equations. Int. Math. Forum, 2: 215-222. 4. Balachandran, K., Sumathy, K. and Kuo, H.H. (2005). "Existence of solutions of general nonlinear Stochastic Voltera Fredholm integral equations". Stochastic Anal. Applic., 23: 827- 851.DOI: 10.1081/SAP-2000644879. 5. Huda, H.O., (2007)." Solutions for the Generalized Multi-Dimensional Voltera Integral and Integro-Differential Equations". M.Sc. Thesis, College of Education, for pure science Ibn Al-Haitham, Baghdad University. 6. Krishnan V. (2006). "probability and random processes " A John Wiley & Sons , Inc., Publication, 7. Milton, J.S., Padgett ,W.J. and Tsokos, C.P. (1972). "On the existence and uniqueness of a random solution to a perturbed random integral equation of the Fredholm type". SIAM J. Applied Math., 22: 194-208. DOI: 10.1137/0122022. 8. Rodgers, J. L. and Nicewander, W. A. (1988). "Thirteen ways to look at the correlation coefficient", The American Statistician, 42(1):59–66, February. 9. Vahidi, A.R. and Mokhtari, M. (2008). "The Decomposition Method for system of linear Fredholm integral equations of the second kind". Applied Math. Sci., 2: 57-62. 10. Vladimir K.K. and Dimitrina, S.D. (2009). "Gamma process pricing path –Dependent Option ", Management Science, 55(3): pp 483 496. 11. Wahdan, M. (2011). “Some Probability Characteristic Functions of the Solution of a Stochastic Linear Fredholm Integral Equation of the Second Kind”, Baghdad Science Journal,8(2): 383 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 بعض المزایا االحصائیھ باالعتماد على اكبر تباین لحل معادلھ فریدھولم التكاملیھ گاماالعشوائیھ ذات البعد الثاني الحاویھ على عملیات محمد وھدان مفلح نور عبد االمیر جبار جامعھ بغداد )/ابن الھیثم(كلیھ التربیھ للعلوم الصرفھ / قسم الریاضیات 2013كانون االول 4، قبل البحث : 2013االول تشرین 9أستلم البحث : الخالصھ معادلة فریدھولم التكاملیة العشوائیة ذات البعد الثاني المتضمنة عملیات گاما المختلفة في ھذا البحث ،تم ایجاد حلین ل تیجة للحلول، اشتقت دوال بالنسبة للمعلمات في حالتین والمتساویة في حالة ثالثة باستخدام طریقة ادومیان التحلیلیة وكن باالعتماد على اكبر التباینات لكل دالة كثافة احتمالیة بالنسبة الى الحاالت t لھا عند الزمن الكثافة االحتمالیھ والتباینات –الثالثة. اشتقت كل من التباین المشترك الذاتي ودوال الكثافة الطیفیة باالضافة الى ذالك احتسبت معامالت االرتباط .كمؤشر لبیان افضلیة الحاالت الثالث ، طریقھ ادومیان التحلیلیھگامالتكاملیھ العشوائیھ ذات البعد الثاني، عملیھ الكلمات المفتاحیھ: معادلھ فریدھولم ا 384 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Some Statistical Characteristics Depending on the Maximum Variance of Solution of Two Dimensional Stochastic Fredholm Integral Equation contains Two Gamma Processes Mohammad W. Muflih Noor A. A. Jabbar Introduction 1. Preliminary So, for m = 0, by (6) and (7): ,,,𝑌-11.,𝑡,𝑤.=,0-∞-,𝑒-−,𝑠+𝑤𝑡..,,𝑌-10.,𝑡,𝑠.+,𝑌-20.,𝑡,𝑠. .𝑑𝑠.-,𝑌-21.,𝑡,𝑤.=,0-∞-,𝑒-−,𝑠+𝑤,𝑡-2..., ,𝑌-10.,𝑡,𝑠.+,𝑌-20.,𝑡,𝑠..𝑑𝑠... ,,,𝑌-11.,𝑡,𝑤.=,0-∞-,𝑒-−,𝑠+𝑤𝑡.., ,,,,𝛽-1..-,𝛼-1.𝑡.-Г,,𝛼-1.𝑡. . ,𝑠-,𝛼-1.𝑡−1. ,𝑒-−,𝛽-1.𝑠.+,,,,𝛽-2..-,𝛼-2.𝑡.-Г,,𝛼-2.𝑡. . ,𝑠-,𝛼-2.𝑡−1. ,𝑒-−,𝛽-2.𝑠..𝑑𝑠.-,𝑌-21.,𝑡,𝑤.=,0-∞-,𝑒-−,𝑠+𝑤,𝑡-2..., ,,,,𝛽-1..-,𝛼-1.𝑡.-Г,,𝛼-1.𝑡... Hence where 1.5 Power Spectral Density Function