On the Joint Distribution of Two Continuous Independent Random Variables

2021 ◽  
Vol 10 (3) ◽  
pp. 641-646
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables

scholarly journals On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

scholarly journals Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes

1991 ◽  
Vol 28 (04) ◽  
pp. 930-933
Author(s):  
C. Y. Teresa Lam
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Renewal Processes ◽  
Life Distribution ◽  
Recurrence Times ◽  
Joint Simulation ◽  
Total Life

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

Joint simulation of backward and forward recurrence times in a renewal process

1989 ◽  
Vol 26 (02) ◽  
pp. 404-407 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Renewal Process ◽  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Recurrence Times ◽  
Joint Simulation

It is shown that, in a renewal process with inter-arrival distributionF,an observation from the asymptotic (whent→∞) joint distribution of backward and forward recurrence times attcan be simulated by simulating an observation of the pair (UW, (1 –U)W), whereUandWare independent random variables withU~ uniform(0, 1) andWdistributed according to the length-biased version ofF.

On the Reliability for Some Bivariate Dependent Beta and Kumaraswamy Distributions: A Brief Survey

2019 ◽  
Vol 34 (2) ◽  
pp. 115-121 ◽  
Author(s):  
Indranil Ghosh
Keyword(s):  
Joint Distribution ◽  
Special Functions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Bivariate Distributions ◽  
Algebraic Form ◽  
Strength Models ◽  
Explicit Expressions

Abstract In the area of stress-strength models, there has been a large amount of work regarding the estimation of the reliability {R=\Pr(X<Y)} . The algebraic form for {R=\Pr(X<Y)} has been worked out for the vast majority of the well-known distributions when X and Y are independent random variables belonging to the same univariate family. In this paper, forms of R are considered when {(X,Y)} follow bivariate distributions with dependence between X and Y. In particular, explicit expressions for R are derived when the joint distribution are dependent bivariate beta and bivariate Kumaraswamy. The calculations involve the use of special functions.

scholarly journals Reliability for some bivariate exponential distributions

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Saralees Nadarajah ◽  
Samuel Kotz
Keyword(s):  
Joint Distribution ◽  
Special Functions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Distributions ◽  
Bivariate Distributions ◽  
Algebraic Form ◽  
Bivariate Exponential ◽  
Strength Models ◽  
Explicit Expressions

In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliabilityR=Pr⁡(X<Y). The algebraic form forR=Pr⁡(X<Y)has been worked out for the vast majority of the well-known distributions whenXandYare independent random variables belonging to the same univariate family. In this paper, forms ofRare considered when(X,Y)follow bivariate distributions with dependence betweenXandY. In particular, explicit expressions forRare derived when the joint distribution isbivariate exponential. The calculations involve the use of special functions. An application of the results is also provided.

Characterizations via conditional distributions

1977 ◽  
Vol 14 (04) ◽  
pp. 806-816
Author(s):  
Robert H. Berk
Keyword(s):  
Order Statistics ◽  
Conditional Distribution ◽  
Joint Distribution ◽  
Exponential Family ◽  
Random Variables ◽  
Independent Random Variables ◽  
Conditional Distributions

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.

scholarly journals A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1505
Author(s):  
Evgeniy Anatolievich Savinov
Keyword(s):  
Absolute Continuity ◽  
Joint Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Necessary Condition ◽  
Conditional Distributions ◽  
One Dimensional ◽  
The Absolute

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension n>1 this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions.

scholarly journals Reliability for some bivariate gamma distributions

2005 ◽  
Vol 2005 (2) ◽  
pp. 151-163 ◽  
Author(s):  
Saralees Nadarajah
Keyword(s):  
Joint Distribution ◽  
Special Functions ◽  
Random Variables ◽  
Bivariate Distribution ◽  
Independent Random Variables ◽  
Algebraic Form ◽  
Gamma Distributions ◽  
Strength Models ◽  
Explicit Expressions

In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliabilityR=Pr(X<Y). The algebraic form forR=Pr(X<Y)has been worked out for the vast majority of the well-known distributions whenXandYare independent random variables belonging to the same univariate family. In this paper, we consider forms ofRwhen(X,Y)follows a bivariate distribution with dependence betweenXandY. In particular, we derive explicit expressions forRwhen the joint distribution is bivariate gamma. The calculations involve the use of special functions.

scholarly journals Reliability for some bivariate beta distributions

2005 ◽  
Vol 2005 (1) ◽  
pp. 101-111 ◽  
Author(s):  
Saralees Nadarajah
Keyword(s):  
Joint Distribution ◽  
Special Functions ◽  
Random Variables ◽  
Bivariate Distribution ◽  
Independent Random Variables ◽  
Algebraic Form ◽  
Beta Distributions ◽  
Strength Models ◽  
Explicit Expressions

In the area of stress-strength models there has been a large amount of work as regards estimation of the reliabilityR=Pr(X<Y). The algebraic form forR=Pr(X<Y)has been worked out for the vast majority of the well-known distributions whenXandYare independent random variables belonging to the same univariate family. In this paper, we consider forms ofRwhen(X,Y)follows a bivariate distribution with dependence betweenXandY. In particular, we derive explicit expressions forRwhen the joint distribution is bivariate beta. The calculations involve the use of special functions.

scholarly journals Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes

1991 ◽  
Vol 28 (4) ◽  
pp. 930-933 ◽  
Author(s):  
C. Y. Teresa Lam
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Renewal Processes ◽  
Life Distribution ◽  
Recurrence Times ◽  
Joint Simulation ◽  
Total Life

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

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