22 Dist of Func RV

# 22 Dist of Func RV - The distribution of a function of a...

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9/7/10 The distribution of a function Suppose we have a random variable X with distribution (PDF) fX(x). We create a new random variable Y as a function of X. That is Y=h(X). The question is, how can we find the distribution of Y? We will denote the PDF of Y as gY(y).

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9/7/10 The Discrete Case: One-to- When X (and thus Y) are discrete, the procedure is fairly straightforward. We basically match events. The function h() is one to one The easiest case is when each unique value of X maps to a unique value of Y as in the case below. In such a case the function h() can be ( 29 ( 29 ) ( 1 y h X P y Y P - = = =
9/7/10 Example Suppose the distribution of X is: Now suppose that Y = h(X) = 10X P(Y=y) = P(X=h-1(y)) X -2 -1 0 1 2 3 P(x) 0.1 0.3 0.1 0.05 0.1 0.35 X=log10(y ) -2 -1 0 1 2 3 y=10x 0.01 0.1 1 10 100 1000 P(Y=y) 0.1 0.3 0.1 0.05 0.1 0.35

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9/7/10 More complex case: Many-to- The function may not always be one to one. Suppose the function is many to one.
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22 Dist of Func RV - The distribution of a function of a...

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