In this exercise, we prove the normalization condition (1.48
Question and Solution
In this exercise, we prove the normalization condition (1.48) for the univariate Gaussian. To do this consider, the inte
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In this exercise, we prove the normalization condition (1.48) for the univariate Gaussian. To do this consider, the integral I = ? ?? exp ? 1 2?2 x2 dx (1.124) which we can evaluate by first writing its square in the form I2 = ? ?? ? ?? exp ? 1 2?2 x2 ? 1 2?2 y2 dx dy. (1.125) Now make the transformation from Cartesian coordinates (x, y) to polar coordinates (r, ?) and then substitute u = r2. Show that, by performing the integrals over ? and u, and then taking the square root of both sides, we obtain I = 2??21/2 . (1.126) Finally, use this result to show that the Gaussian distribution N (x|
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