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M5L27 - M5L9 Functions of Multiple Random Variables-2 1...

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    M5L9 Functions of Multiple Random Variables-2 1. Introduction In this lecture, various statistical properties of one function of two random variables are discussed. The sum and difference of independent Normal variates, maximum, minimum product and quotient of two random variables are discussed in detail along with numerical examples. 2. Sum and Difference of Normal Variates If and are independent normal variates, having mean as X , Y and standard deviation as X , Y respectively.  Y X Z is one function for these two variables and its pdf can be expressed as,     dy y f y z f z f Y X Z Now this equation for pdf of Z can be expanded as, dy by ay exp z exp dy y y z exp z f X X Y Y Y X Y Y X X Y X Z 2 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2  where, 2 2 2 2 1 1 Y Y X X Y X z b and a Now, considering the last integral part of the expression for pdf and taking a substitution , we get, a b exp a dw aw exp e dy by ay exp a / b 2 2 2 1 2 2 1 2 2 2 2 2
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    Thus we obtain,   2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 2 1 Y X Y X Y X X Y X X Y Y Y X Y X X X Y Y Y X Z z z exp a b exp a z exp z f    2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 1 2 1 Y X Y X Y X Y X Y X Y X Y X X Y X Y Y X z exp z z exp     So, the sum of two independent Gaussian (Normal) random variates also possesses a Gaussian (Normal) distribution with mean and variance as Y X Z and 2 2 2 Y X Z respectively.
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M5L27 - M5L9 Functions of Multiple Random Variables-2 1...

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