Consider a probability density px(x) defined over a continuo
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Consider a probability density px(x) defined over a continuous variable x, and suppose that we make a nonlinear change o
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Consider a probability density px(x) defined over a continuous variable x, and suppose that we make a nonlinear change of variable using x = g(y), so that the density transforms according to (1.27). By differentiating (1.27), show that the location y of the maximum of the density in y is not in general related to the location x of the maximum of the density over x by the simple functional relation x = g(y) as a consequence of the Jacobian factor. This shows that the maximum of a probability density (in contrast to a simple function) is dependent on the choice of variable. Verify that, in the case of a linear transformation, the location of the maximum transforms in the same way as the variable itself.
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