7.3 Chapter Summary

This chapter focused on methods for simplifying sums and integrals. This chapter can be summarized with 4 main techniques:

1. Sums and integrals of linear functions are linear functions of sums and integrals (you can move linear functions outside a sum or an integral),

2. Integration/summation by parts,

3. Integration by substitution/change of variables, and

4. Change of density.

Interactive Tools:¶

1. Change of Density Visualizer - Use this tool to visualize changes of density.

Sums and Integrals of Sums and Products¶

Definitions and examples are all available in Section 7.1.

1. Sums and integrals are linear so, given $f(x) = a g(x) + b h(x)$ then:

   $$\begin{aligned} &\sum_{\text{all } x} f(x) = a \sum_{\text{all } x} g(x) + b \sum_{\text{all } x} h(x), \\ &\int_{\text{all } x} f(x) dx = a \int_{\text{all } x} g(x) dx + b \int_{\text{all } x} h(x) dx. \end{aligned}$$

   (1)

2. Integration by Parts:

   Given $f(x) = g(x) h'(x)$,

   $$\int_{x = a}^b f(x) dx = g(x) h(x) \Big|_{x = a}^b - \int_{a}^b g'(x) h(x) dx$$

   (2)

   where $h'(x) = \frac{d}{dx} h(x)$ and $g'(x) = \frac{d}{dx} g(x)$.

3. Tail Integrals/Sums:

   By integration/summation by parts, if $X$ is nonnegative random variable, then:

   $$\mathbb{E}[X] = \begin{cases} \int_{x = 0}^\infty \text{Pr}(X > x) dx & \text{ if continuous} \\ \sum_{x = 1}^{\infty} \text{Pr}(X \geq x) & \text{ if discrete} \end{cases}$$

   (3)

Change of Variables:¶

Definitions and examples are all available in Section 7.2.

1. To integrate by change of variables (“$u$-substitution”):

   Suppose that $f(x) = g \circ h(x) = g(h(x))$ and that $h(x)$ is a monotonic function. Let $u = h(x)$ and $du(x) = h'(x) dx$.

   Let $h^{-1}$ denote the inverse function that recovers $x$ from $u$, $x = h^{-1}(u)$.

   Then:

   $$\int g(h(x)) dx = \int \frac{g(u)}{h'(h^{-1}(u))} du = \int \tilde{g}(u) du.$$

   (4)

   Let $\tilde{G}$ denote the indefinite integral of $\tilde{g}(x) = g(u)/h'(h^{-1}(u))$. Then:

   $$\int_{a}^b g(h(x)) dx = \tilde{G}(u) \Big|_{h(a)}^{h(b)} = \tilde{G}(h(b)) - \tilde{G}(h(a)).$$

   (5)

2. Applying the change of variables formulas to probability densities produces the change of density formula:

   If $X$ is a continuous random variable with density $f_X$, and $Y = h(X)$ for some differentiable, monotonic function $h$, then:

   $$f_Y(y) = f_X(x) \frac{1}{|h'(x)|} \text{ at } x = h^{-1}(y).$$

   (6)

   If $h(x) = \sigma x + s$ for some $\sigma > 0$, then we can use the linear change of density formula:

   $$f_Y(y) \propto f_X(h^{-1}(y)) = f_X\left( \frac{x - s}{\sigma} \right).$$

   (7)