# Bishop - Pattern Recognition & Machine Learning, Exercise 1.4

I'm working on exercise 1.4 in Bishops Pattern Recognition & Machine Learning book.

At first I don't understand equation 1.27. He writes: "Under a nonlinear change of variable , a probability density transforms differently from a simple function, due to the Jacobian Factor."

I never ever heard about the Jacobian factor. What is that factor?

"For instance, if we consider a change of variables $$x = g(y)$$, then a function $$f(x)$$ becomes $$\tilde f(g(y))$$. Now consider a probabilty density $$p_x(x)$$ that corresponds to a density $$p_y(y)$$ with respect to the new variable $$y$$, where the suffices denote the fact that $$p_x(x)$$ and $$p_y(y)$$ are different densities. Observations falling in range $$(x, x + \delta x)$$ will, for small values of $$\delta x$$, be transformed into the range $$(y, \delta y)$$ where $$p_x(x)\delta x \simeq p_y(y)\delta y$$, [...]"

What does the relation $$\simeq$$ mean in this context?

"[...] and hence \begin{align} p_y(y) &= p_x(x) \left| \frac{\text{d}x}{\text{d}y}\right|\\ &= p_x(g(y))\left|g'(y)\right|." \end{align}

This is equation 1.27. I don't understand where this equation comes from. Why is there this absolute value?

"One consquence of this property is that the concept of the maximum of a probabilty density is dependent on the choice of variable."

And at this point the book refers to exercise 1.4:

"Consider a probability density $$p_x(x)$$ defined over a continous variable $$x$$, and suppose that we make a nonlinear change of variable using $$x = g(y)$$, so that the density transforms according (1.27). By differentiating (1.27), show that the location $$\hat y$$ of the maximum of the density in $$y$$ is not in general related to the location $$\hat x$$ of the maximum of the density over $$x$$ by the simple functional relation $$\hat x = g(\hat y)$$ 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."

I don't understand, what this exercise asks me to do... :/

Would be great, if someone could help me...