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**Unformatted text preview: **p X then Y = T ( X ) is also discrete. The pmf of Y : p Y ( y j ) = P ( Y = y j ) = P ( T ( X ) = y j ) = X x i : T ( x i )= y j p X ( x i ) . If the function T is one-to-one , then p Y ( y j ) = p X ± T-1 ( y j ) ² ( T-1 is the inverse map ) Example Let X be a mean λ Poisson random variable. Find the distribution of Y = 2 X . Let X be continuous with pdf f X . • Sometimes Y = T ( X ) is also continuous. • To compute the density of Y adjust by the derivative of the inverse transfor-mation . Monotone transformations : the function T is either increasing or decreasing on the range of X . A monotone function T is automatically one-to-one. The pdf of Y : f Y ( y ) = f X ± T-1 ( y ) ² ³ ³ ³ ³ ³ dT-1 ( y ) dy ³ ³ ³ ³ ³ ....

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- '10
- SAMORODNITSKY
- Probability theory, Probability mass function