The Distributions of Sums, Products and Ratios of Inverted Bivariate Beta Distribution 1

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1 Applied Mathematical Sciences, Vol. 2, 28, no. 48, The Distributions of Sums, Products and Ratios of Inverted Bivariate Beta Distribution 1 A. S. Al-Ruzaiza and Awad El-Gohary 2 Department of Statistics and O.R., College of Science King Saud University, P.O. Box 2455 Riyadh 11451, Saudi Arabia aigohary@ksu.edu.sa Abstract In this paper we derive the exact distributions of sums, products, and ratios of two random variables when they follow the bivariate inverted beta distribution. Forms of the probability density functions of these distributions are presented. The moments of theses distributions are derived. We provide extensive tabulation of the percentiles points associated with the distributions obtained. Keywords: Bivariate inverted beta distributions, Sums, Ratios, Percentiles, Moments 1 Introduction The statistics literature has seen many developments in the theory and applications of linear combinations, ratios and products of random variables. Nadarajah and Gupta have derived the distributions of sums and ratios of beta Stacy distribution see [1]. Olkin and Liu have presented a bivariate beta distribution that has support on the simplex x i 1, (i =1, 2). This distribution has a positively likelihood ratio dependent and hence it has positive quadratic dependent [2]. El-Gohary and Sarhan have derived distributions for the sums, products, ratios and differences of Marshall-Olkin bivariate exponential distribution [4]. El-Gohary has obtained a new class of multivariate 1 This research was supported by the College of Science Research Center at King Saud University under project No. Stat/28/39. 2 Permanent address: Department of Mathematics, College of Science, Mansoura University, Mansoura 35516, Egypt

2 2378 A. S. Al-Ruzaiza and A. El-Gohary linear failure rate distribution of Marshall-Olkin type [5] and has discussed its applications in the reliability theory, see [7, 8]. This paper produces the distributions of the sums, rations and products of two dependent random variables follow a bivariate inverted beta distribution. Such these distribution are interesting in statistics and several applications, such as reliability engineering, industrial engineering and computer systems. The Dirichlet distribution is often used as a prior distribution for the parameters of a multinormal distribution. Univariate beta distributions are used extensively use in Bayesan statistics, since beta distributions provide a family of conjugate prior distributions for binomial distributions. Many applications in the mathematical, physical and engineering sciences, one random variable is functionally related to two or more different random variables. For example, the random signal S at the input of an amplifier consists of the sum of a random signal to which is add independent random noise. Therefore the random signal is a sum of two random variables. If X is a random variable what is the probability density function of S. Many signal processing systems use electronic multipliers to multiply two signals together. If X is the signal on one input and Y is the signal on the other input, what is the probability density function of the output P = XY. Linear combination of two random variables is one of interesting problems in automation, control, computer science and related fields. The ratio of two random variables takes place in the stress-strength model in the context of reliability. For example, the model describes the lifetime of a component that has a random strength X and subjected to random stress Y. The components fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever X > Y. Therefore, P (Y > X) =P (2Y/(X + Y ) < 1) is a measure of components reliability. In this paper we will derive the exact distributions of S = X + Y, D = X Y, P = XY and R = X/(X + Y ) and their moments when the random variables X and Y are correlated inverted bivariate beta distribution with the joint probability density function given by [3] f(x, y) = Γ(θ 1 + θ 2 + θ 3 ) x θ1 1 y θ 2 1 Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) (1 + x + y) θ 1+θ 2 +θ 3,x>,y >,θ i >,i=1, 2.3 (1.1) Also, moments for the distributions of sums, products and ratios will be derived. Further, we provide extensive tabulations of the associated percentage points, obtained by intensive computer power. This paper is organized as follows. Section 2 presents the probability density functions and cumulative distribution functions of S = X + Y, P = XY

3 Distributions of sums, products and ratios 2379 and R = X/(X + Y ). The moments of S = X + Y, P = XY and R = X/(X + Y ) are given in Section 3. Finally, in Section 4, the tabulations of the percentiles of P are presented. The calculations through out this paper involve the complete beta function and Euler hypergeometric function that defined by u a 1 B(a, b) = du, (1.2) (1 + u) a+b 2F 1 ( μ, λ;2μ;1 k 2 /k 1 ) = for 4μ >λ>. and k λ 1 B(λ, 2μ λ) u λ 1 du (1+(k 1 + k 2 )u + k 1 k 2 u 2 ) μ, (1.3) x 2F 1 (a, 1 b; a +1;x) =ax a u a 1 (1 u) b 1 du, (1.4) pf q (α 1,α 2,...,α p,β 1,β 2,...,β p )= k= (α 1 ) k, (α 2 ) k...,(α p ) k z k (β 1 ) k, (β 2 ) k,...(β q ) k!k (1.5) where (α i ) k = α i (α i + 1)(α i +2)...(α i + k 1) is called a generalized hypergeometric series. For the full properties of these special functions, we refer to Gradshteyn and Ryzhik (2) [9]. 2 Probability density functions In this section, we derive the exact probability density functions and cumulative distribution functions of the sums, products, ratios and differences of two random variables that follow from Inverted bivariate beta distribution. The following theorem give the probability density function for S = X + Y in exact form when the random variables X and Y are distributed according to (1.1) Theorem 2.1 If X and Y are jointly distributed according to (1.1), then the probability density function and cumulative distribution function of S = X + Y are given, respectively, by s θ 1+θ 2 1 f S (s) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 + θ 2 )Γ(θ 3 ) (1 + s) θ 1+θ 2 +θ 3,s>. (2.1)

4 238 A. S. Al-Ruzaiza and A. El-Gohary and F S (s) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 + θ 2 + 1)Γ(θ 3 ) sθ 1+θ F 1 (θ 1 + θ 2 + θ 3,θ 1 + θ 2 ; θ 1 + θ 2 +1;s),s> (2.2) Proof. The proof of this theorem can be achieved Firstly, by deriving the joint pdf of (S, R) =(X + Y, X/(X + Y )). Using the joint pdf of (X,Y), given by (1.1), we can get the joint pdf of (S, R) as g(s, r) = Γ(θ 1 + θ 2 + θ 3 ) r θ1 1 (1 r) θ2 1 s θ 1+θ 2 1 Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) (1 + s) θ 1+θ 2 +θ 3,s>, <r<1 (2.3) Therefore, the marginal pdf of S can be derived from g(s, r) as follows s θ 1+θ 2 1 f S (s) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) (1 + s) θ 1+θ 2 +θ 3 1 r θ 1 1 (1 r) θ 2 1 dr Solving the integrals in the above relations using Gradshteyn and Ryzhik (2) the results obtained, one gets f S (s) as given in (2.1). Using the well known relation between pdf and cdf and (2.1), one can derive the cdf of s as given by (2.2), which completes the proof. The distribution of sums of two correlated random variables has many several applications. For example, the distribution of the sums of two random variables gives the distribution of income of a person who has two different covariant incomes. Plots of the sums density function (2.1) for some selected values of distribution parameters are shown in figure 1. Fig. 1a pdf of the sums S=X+Y Fig. 1b pdf of the sums S=X+Y f(s) f(s) infinity infinity s s Figures 1a and 1b display the graph of pdf of S at θ 1 =1.5, θ 2 =2,θ =.5, and θ 1 =5.5, θ 2 =2,θ 3 =2.5 respectively.

5 Distributions of sums, products and ratios 2381 Fig 1c pdf of the sums S=X+Y infinity Fig 1d pdf of the sums S=X+Y f(s) f(s) infinity infinity s s Figures 1c and 1d display the graph of pdf of S at θ 1 =1.5, θ 2 =25,θ 3 = 5.5, and θ 1 =15.5, θ 2 =5,θ 3 =2.5 respectively. Theorem 2.2 If X and Y are jointly distributed according to (1.1), the probability density function and cumulative distribution function of the rations R = X/S are given, respectively, by f R (r) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) B(θ 1 + θ 2,θ 3 2) r θ 1 1 (1 r) θ 2,θ 3 > 2, <r<1 and F R (r) = Γ(θ 1 + θ 2 + θ 3 )B(θ 1 + θ 2,θ 3 2) Γ(θ 1 + 1)Γ(θ 2 Γ(θ 3 ) Proof. From (2.3), we have f R (r) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) rθ 1 1 (1 r) θ 2 1 (2.4) r θ 1 2 F 1 (θ 1, θ 2 ; θ 1 +1;r), θ 3 > 2, r<1 (2.5) s θ 1+θ 2 1 (1 + s) θ 1+θ 2 +θ 3 ds (2.6) Solving the integrals in (2.6), using (1.2) one gets f R (r) as given in (2.4). Then, using (2.4) together with the well known relation between the pdf and cdf, one gets F R as given by (2.5), which completes the proof of the theorem. Plots of the sums density function (2.4) for some selected values of distribution parameters are shown in figures 2. Figure 2 displays the pdf of R = X/(X + Y ), at different sets of the parameters θ 1,θ 2 and θ.

6 2382 A. S. Al-Ruzaiza and A. El-Gohary 18 Fig 2a pdf of the ratios R=X/(X+Y) Fig 2b pdf of the ratios R=X/(X+Y) f(s) f(s) s s Figures 2a and 2b display the graph of pdf of S at θ 1 =5,θ 2 =15,θ 2 =2.5, and θ 1 = θ 2 = θ = 1 respectively. Fig 2c pdf of the ratios R=X/(X+Y) Fig 2d pdf of the ratios R=X/(X+Y) f(r) 4 f(r) r r Figures 2c and 2d display the graph of pdf of S at θ 1 =1.5, θ 2 =5,θ 3 =5.5, and θ 1 =15,θ 2 =5.5, θ 3 = 2 respectively. Theorem 2.3 If X and Y are jointly distributed according to (1.1), then the probability density function of the products P = XY takes the following form f P (p) = 22θ 1+θ 3 Γ(θ 1 + θ 2 + θ 3 )B(2θ 1 + θ 3, 2θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 )(1+ p θ p) 2θ 1+θ 3 2 2F 1 (θ 1 4p ) 1 + θ 2 + θ 3, 2θ 1 + θ 3 ;2θ 1 +2θ 2 +2θ 3, 1+,p>.(2.7) 1 4p Proof. Starting with the joint pdf of (X, P) =(X, XY ) as in the following form x 2θ 1+θ 3 1 p θ 2 1 g(x, p) = Γ(θ 1 + θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) (p + x + x 2 ) θ,x>,p >,θ 1+θ 2 +θ 3 i > (i =1, 2, 3)(2.8)

7 Distributions of sums, products and ratios 2383 Therefore, the marginal pdf of P can be derived from g(x, p) as follows f P (p) = g(x, p) dx (2.9) Substituting from the above relations (2.8) into (2.9) and using the relation (1.3), one gets f P (p) as given by (2.7), which completes the proof of the theorem. Next, we derive exacts forms for the moments of sums, rations and products. 3 Moments of sums, ratios and products In this section we derive the moments of S = X + Y, P = XY and R = X/(X +Y ) when X and Y are distributed according to (1.1). Now we establish the following lemma. Lemma 3.1 If X and Y are jointly distributed according to (1.1) then E(X n Y m )= Γ(θ 1 + θ 2 + θ) ( ) ( ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) B θ 1 + m, θ 2 + θ 3 m B θ 2 + n, θ 3 m n,θ 3 >m+ n Proof. Starting with the following well known relation E(X n Y m )= Substituting from (1.1) into (3.2), one gets x n y m f(x, y)dxdy (3.2) E(X n Y m )= Γ(θ 1 + θ 2 + θ 3 ) x θ1+m 1 y θ+n 1 Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) (1 + x + y) θ 1+θ 2 +θ 3 dx dy. (3.3) Using the definition of the beta function (1.3) and after some algebraic manipulation one gets the formula (3.1) of the moments. Theorem 3.1 If X and Y are jointly distributed according to (1.1), then E(S m )= Γ(θ 1 + θ 2 + θ 3 )Γ(θ 1 + θ 2 + θ 3 2m) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 )Γ(θ 1 m)γ(θ 2 + θ 2 m) for θ 3 > 2m k. m B(θ 2 + m k, θ 3 + k 2m), (3.4) Proof. Starting with the definition of expectation and using the binomial expansion, one gets m ( ) m E(S m )=E((X + Y ) m )= E(X k Y m k ) k k= k= (3.1)

8 2384 A. S. Al-Ruzaiza and A. El-Gohary Applying lemma (3.1), we get (3.5), which completes the proof of the lemma.. Theorem 3.2 If X and Y are jointly distributed according to (1.1), then E(P m )= Γ( θ 1 + θ 2 + θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) B( θ 1 + m, θ 2 + θ 3 m ) B ( θ 2 + m, θ 3 2m ),θ 3 > 2m.(3.5) Proof. Starting by setting m = n in the relation (3.1) we get the formula (3.5) which completes the proof.. Theorem 3.3 If X and Y are jointly distributed according to (1.1), then E(R m )= Γ(θ 1 + θ 2 + θ 3 )Γ(θ 1 + θ 2 + θ 3 2) Γ(θ 1 )Γ(θ 2 Γ(θ 3 )Γ(θ 1 + θ 2 )Γ(θ 3 2) B( θ 2 + m, θ 2 +1 ),θ 3 > 2 (3.6) for n 1. Proof. Using (2.4), one gets E(R n )= Γ(θ 1 + θ 2 + θ 3 )Γ(θ 1 + θ 2 + θ 3 2) Γ(θ 1 )Γ(θ 2 Γ(θ 3 )Γ(θ 1 + θ 2 )Γ(θ 3 2) 1 r θ 1+m 1 (1 r) θ 2 dr, (3.7) Using the complete beta function definition yields (3.7), which completes the proof.. 4 Percentiles points for sums, ratios and products In this section, we provide extensive tabulations of the percentiles of the distributions of S, R and P. This percentiles are computed numerically by solving the following equations, with respect to s q,r q and p q respectively, sq f S (t)dt = q, rq f R (t)dt = q, pq f P (p)dp = q (4.1) where the probability density function f S (s), f R (r) and f P (p) are given by (2.1), (2.6) and (2.7) respectively. In fact this function involves computation of Euler hypergeometric, gamma and functions and routines for these are widely available.

9 Distributions of sums, products and ratios 2385 In the other hand the percentiles of the distributions of S, and R can be easily computed by using the expressions for their cumulative distribution functions F S (s), and F R (r), respectively. Inverting the equations F S (s q )= q, F R (r) =q, F p (p q )=q, one gets. The percentage point for the sums S q = X +Y distribution can be obtained by solving the equation, with respect to s q, Γ ( ) θ 1 + θ 2 + θ 3 Γ ( ) ( )s θ 1+θ 2 ( ) +1 θ 1 + θ 2 Γ q 2F 1 θ1 + θ 2 + θ 3,θ 1 + θ 2 ; θ 1 + θ 2 + θ 3 +1,s q = q θ3 (4.2) The percentage point for the ratios R q = X/(X + Y ) distribution can be obtained by solving the equation, with respect to r q, Γ ( θ 1 + θ 2 + θ 3 ) B(θ1 + θ 2,θ 3 2) Γ ( θ 1 +1 ) Γ ( θ 2 ) Γ ( θ3 ) r θ 1 q 2F 1 ( θ1, θ 2 ; θ 1 +1,r q ) = q (4.3) Finally, percentage points for P q = XY can be obtained by solving the equation, with respect to p q, 2 2θ 1+θ 3 Γ(θ 1 + θ 2 + θ 3 ) pq { t θ 1 1 B(2θ 1 +θ 3, 2θ 2 +θ 3 ) Γ(θ 1 )Γ(θ 2 )Γ(θ 3 ) ( t) 2θ 1+θ 3 2F 1 (θ 1 + θ 2 + θ 3, 2θ 1 + θ 3 ;2θ 1 +2θ 2 +2θ 3, 2 1 4t ) } 1+ dt = q, 1 4t (4.4) The following tables give the numerical solution (4.2), (4.3) and (4.4) for different values of the parameters θ 1,θ 2 and θ 3. Table 1. Displays the numerical values of the percentiles of the sums S = X + Y corresponding the probabilities.9,.95,.975,.99,.995,.999 for different values of the parameters θ 1,θ 2 and θ 3.

10 2386 A. S. Al-Ruzaiza and A. El-Gohary Table 1 q θ 1 θ 2 θ s q Table 2. Displays the numerical values of the percentiles of the ratios R = x/(x + Y ) corresponding the probabilities.9,.95,.975,.99,.995,.999 for different values of the parameters θ 1,θ 2 and θ 3.

11 Distributions of sums, products and ratios 2387 Table 2. q θ 1 θ 2 θ r q

12 2388 A. S. Al-Ruzaiza and A. El-Gohary q θ 1 θ 2 θ

13 Distributions of sums, products and ratios 2389 Table 3. Displays the numerical values of the percentiles of the products P = XY corresponding the probabilities.9,.95,.975,.99,.995,.999 for different values of the parameters θ 1,θ 2 and θ 3. Table 3 q θ 1 θ 2 θ p q

14 239 A. S. Al-Ruzaiza and A. El-Gohary Table 3 continued q θ 1 θ 2 θ

15 Distributions of sums, products and ratios Conclusion Finally, we conclude that the distributions of sums, ratios, and products of the bivariate distribution of X and Y follow from bivariate inverted beta distribution are obtained. Also the moments of S = X + Y, R = X/(X + Y ) and P = XY are derived. Tabulation for the percentage points of, sums, ratios and products are derived. References [1] Gupta, A. and Ndarajah, S. (26), Sums and ratios for beta Stacy distribution, Applied Mathematics and Computation 173, pp [2] Olkin, I. and Liu, R. (23), A bivariate distribution, it Statistics & Probability Letters,pp [3] Zografos, K. and Ndarajah, S. (25), Expressions for Renyi and Shannon entropies for multivariate distributions, it Statistics & Probability Letters,pp [4] El-Gohary, A. and Sarhan, A. (26), The Distributions of Sums, Products, Ratios and Differences for Marshall-Olkin Bivariate Exponential Distribution, International Journal of Applied Mathematics, In Press. [5] Marshall, A. W. and Olkin, I. A. (1967b). A generalized bivariate exponential distribution, J. Appl. Prob., 4: [6] Lee, L. (1979), Multivariate distributions having Weibull properies. Journal of Multivariate analysis, 9, pp [7] El-Gohary, A. Bayes estimation of parameters in a three non-independent components series system with time dependent failure rate, Applied Mathematics and Computation, Vol. 158, Issue 1, 25 October 24, pp [8] El-Gohary, A. (25) A multivariate mixture of linear failure rate distribution in reliability models, International Journal of Reliability and Applications, In Press. [9] Gradshteyn, I. S. and Ryzhih, I. M. Table of integrals, Series and Products, Academic press New York (2). Received: March 21, 28