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Probability: Difference between revisions
→Change of variables
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===Change of variables=== | ===Change of variables=== | ||
Let <math>g</math> be a monotonic increasing function and <math>Y = g(X)</math>. | Let <math>g</math> be a monotonic increasing function and <math>Y = g(X)</math>.<br> | ||
Then <math>F_Y(y) = P(Y \leq y) = P(X \leq g^{-1}(y)) = F_X(g^{-1}(y))</math>. | Then <math>F_Y(y) = P(Y \leq y) = P(X \leq g^{-1}(y)) = F_X(g^{-1}(y))</math>.<br> | ||
And <math>f_Y(y) = (d/dy)F_Y(y) = (d/dy) F_X(g^{-1}(y)) = f_X(g^{-1}(y)) (d/dy)g^{-1}(y)</math> | And <math>f_Y(y) = (d/dy)F_Y(y) = (d/dy) F_X(g^{-1}(y)) = f_X(g^{-1}(y)) (d/dy)g^{-1}(y)</math><br> | ||
==Expectation and Variance== | ==Expectation and Variance== | ||