Integral Substitution, Jacobians, and Probability Densities

All flashcards

Integration by substitution

A method that changes the integration variable, often reversing the chain rule to simplify an integral.

Definite-integral substitution

For $u=g(x)$, replace the limits $a,b$ with $g(a),g(b)$ when integrating in $u$.

Jacobian determinant

The determinant of a transformation's derivative matrix; its absolute value gives local volume scaling.

One-dimensional density transformation

For an invertible differentiable map $y=\varphi(x)$, $p_Y(y)=p_X(\varphi^{-1}(y))\left|\frac{d\varphi^{-1}}{dy}\right|$.

Trigonometric substitution for $\sqrt{1-x^2}$

Setting $x=\sin u$ changes the radical to $\sqrt{1-\sin^2u}$, which equals $\cos u$ when $\cos u\geq 0$.

Quiz questions

1. For a definite integral with $u=g(x)$, what limits should be used when rewriting the integral in terms of $u$?Answer: The limits $g(a)$ and $g(b)$.
2. What does the absolute value of a Jacobian determinant measure in a change of variables?Answer: Local scaling of area or volume.
3. For an invertible differentiable transformation $y=\varphi(x)$, which expression gives the transformed density?Answer: $p_X(\varphi^{-1}(y))\left|\frac{d\varphi^{-1}}{dy}\right|$
4. Which is an antiderivative of $\tan x$ on an interval where it is defined?Answer: $\ln|\sec x|+C$
5. Why is a change-of-variables map usually required to be one-to-one on the region being transformed?Answer: To avoid counting points in the transformed region multiple times.