PROBABILITY DISTRIBUTION OF SUM OF TWO CONTINUOUS VARIABLES AND CONVOLUTION

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Journal for Research| Volume 02| Issue 01 | March 2016 ISSN: 2395-7549

Probability Distribution of Sum of Two Continuous Variables and Convolution Rashmi R. Keshvani Professor Department of Mathematics Sarvajanik College of Engineering & Technology, Surat (Guj.) India

Yamini M. Parmar Assistant Professor Department of Mathematics Government Engineering College, Gandhi nagar (Guj), India

Abstract All physical subjects, involving random phenomena, something depending upon chance, naturally find their own way to theory of Statistics. Hence there arise relations between the results derived for hose random phenomena in different physical subjects and the concepts of Statistics. Convolution theorem has a variety of applications in field of Fourier transforms and many other situations, but it bears beautiful applications in field of statistics also .Here in this paper authors want to discuss some notions of Electrical Engineering in terms of convolution of some probability distributions. Keywords: A Probability Distribution, An Uniform Probability Distribution, Central Limit Theorem, Convolution, Mean And Variance of a Probability Distribution, Triangular Function, Unit Rectangle Function _______________________________________________________________________________________________________

I. INTRODUCTION Occurrence of resistance of a resistor, with its tolerance can be expressed as a probability distribution. In those circumstances, what would be the resultant distribution describing occurrence of resistances of resistors having different resistance and different tolerances, when combined in series? How means and variances are interrelated with that resultant distribution, is main focus of this paper. In other words, to obtain probability distribution of sum of two random variables is main objective of this paper.

II. SOME BASIC CONCEPTS OF STATISTICS A Probability Distribution, Its Mean and Variance: A real valued function đ?‘?(đ?‘Ľ) is said to be a probability distribution of a random variable đ?‘Ľ, if (1) đ?‘?(đ?‘Ľ) ≼ 0, ∀ đ?‘Ľ ∈ đ?‘… and ∞ ∑đ?‘Žđ?‘™đ?‘™ đ?‘Ľ đ?‘?(đ?‘Ľ) = 1 where đ?‘Ľ is a discrete random variable. or (2) đ?‘?(đ?‘Ľ) ≼ 0, ∀ đ?‘Ľ ∈ đ?‘… and âˆŤâˆ’âˆž đ?‘?(đ?‘Ľ)đ?‘‘đ?‘Ľ = 1 where đ?‘Ľ is a continuous random variable. The mean of probability distribution đ?‘?(đ?‘Ľ), denoted by Îź, is defined as ∞ (1) đ?œ‡ = ∑đ?‘Žđ?‘™đ?‘™ đ?‘Ľ đ?‘Ľ đ?‘?(đ?‘Ľ), if đ?‘Ľ is discrete, or (2) đ?œ‡ = âˆŤâˆ’âˆž đ?‘Ľ đ?‘?(đ?‘Ľ)đ?‘‘đ?‘Ľ, if đ?‘Ľ is continuous. (1) The variance of probability distribution đ?‘?(đ?‘Ľ), denoted by đ?œŽ 2 , is defined as ∞ (1) đ?œŽ 2 = ∑đ?‘Žđ?‘™đ?‘™ đ?‘Ľ ( đ?‘Ľ − đ?œ‡ )2 đ?‘?(đ?‘Ľ), if đ?‘Ľ is discrete, or (2) đ?œŽ 2 = âˆŤâˆ’âˆž( đ?‘Ľ − đ?œ‡ )2 đ?‘?(đ?‘Ľ)đ?‘‘đ?‘Ľ, if đ?‘Ľ is continuous. [1] (2) The Uniform Probability Distribution: The uniform probability distribution, with parameters đ?›ź and đ?›˝, has probability distribution đ?‘?(đ?‘Ľ) defined as 1 đ?‘–đ?‘“ đ?›ź < đ?‘Ľ < đ?›˝ đ?‘?(đ?‘Ľ) = { đ?›˝âˆ’đ?›ź 0 đ?‘’đ?‘™đ?‘ đ?‘’đ?‘¤â„Žđ?‘’đ?‘&#x;đ?‘’

(3)

III. THE CONCEPT OF CONVOLUTION The convolution of two functions đ?‘“(đ?‘Ľ) and đ?‘”(đ?‘Ľ), [2], denoted by đ?‘“(đ?‘Ľ) ∗ đ?‘”(đ?‘Ľ), or (đ?‘“ ∗ đ?‘”)(đ?‘Ľ) is defined as ∞ đ?‘“(đ?‘Ľ) ∗ đ?‘”(đ?‘Ľ) = âˆŤâˆ’âˆž đ?‘“(đ?‘˘)đ?‘”(đ?‘Ľ − đ?‘˘)đ?‘‘đ?‘˘ (4) One can easily check that the operation of convolution is commutative, associative and also distributive over addition. Convolution has additive property also. That is ∞ đ?‘“ ∗ (đ?‘”1 + đ?‘”2 )(đ?‘Ľ) = đ?‘“(đ?‘Ľ) ∗ (đ?‘”1 + đ?‘”2 )(đ?‘Ľ) = đ?‘“(đ?‘Ľ) ∗ (đ?‘”1 (đ?‘Ľ) + đ?‘”2 (đ?‘Ľ)) = âˆŤâˆ’âˆž đ?‘“(đ?‘˘) (đ?‘”1 (đ?‘Ľ − đ?‘˘) + đ?‘”2 ( đ?‘Ľ − đ?‘˘))đ?‘‘đ?‘˘ = −∞ −∞ âˆŤâˆž đ?‘“(đ?‘˘)đ?‘”1 (đ?‘Ľ − đ?‘˘)đ?‘‘đ?‘˘ + âˆŤâˆž đ?‘“(đ?‘˘)đ?‘”2 (đ?‘Ľ − đ?‘˘)đ?‘‘đ?‘˘ = ( đ?‘“ ∗ đ?‘”1 )(đ?‘Ľ) + ( đ?‘“ ∗ đ?‘”2 )(đ?‘Ľ) That is đ?‘“ ∗ (đ?‘”1 + đ?‘”2 ) = ( đ?‘“ ∗ đ?‘”1 ) + ( đ?‘“ ∗ đ?‘”2 )

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