5,336
edits
Probability: Difference between revisions
→Change of variables
| Line 38: | Line 38: | ||
Let <math>g</math> be a monotonic increasing function and <math>Y = g(X)</math>.<br> | 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>.<br> | 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) = | And <math>f_Y(y) = \frac{d}{dy} F_Y(y) = \frac{d}{dy} F_X(g^{-1}(y)) = f_X(g^{-1}(y)) \frac{d}{dy}g^{-1}(y)</math><br> | ||
Hence: | |||
<math display="block"> | |||
f_Y(y) = f_x(g^{-1}(y)) \frac{d}{dy} g^{-1}(y) | |||
</math> | |||
==Expectation and Variance== | ==Expectation and Variance== | ||