Chapter 5b - CH 5 Part 2 Functions of Random Variables...

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CH 5 Part 2 – Functions of Random Variables (Really Sec. 4.8, 5.5) 1
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2 Functions of Random Variables Rules Regarding Expectation/Variance: Let Y = X + c, where X is a random variables with mean μ and variance σ 2 and c is a constant. E(Y) = E(X + c) = E(X) + E(c) = E(X) + c = μ + c Var(Y) = Var(X + c) = Var(X) + Var(c) = Var(X) = σ 2 Let Y=cX. E(Y) = E(cX) = c*E(x) = c μ Var(Y) = Var(cX) = c 2 Var(X) = c 2 σ 2
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How do we find the distribution function for this new variable? Univariate (one variable) case: (Sec. 4.8) Suppose we have one continuous random variable X, and define Y = g(X), where g is strictly monotonic and differentiable. Then f(y) = where g -1 (y) is the function of g(x) solved for x, and d g -1 (y)/dy is the derivative of g(x) w.r.t. y. | ) ( | ))* ( ( 1 1 dy y dg y g f x - - 3
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Univariate Example Let f(x) = 2X, 0 < X < 1. We have found that E(X) = 2/3 and Var(X) = 1/18 . We create Y=g(X)=3X+6. (a)Find the probability distribution for Y. 4
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This note was uploaded on 01/17/2011 for the course STAT 4714 at Virginia Tech.

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Chapter 5b - CH 5 Part 2 Functions of Random Variables...

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