10.4 Chapter Summary

Interactive Tools:¶

1. Probability as Volume Visualizer - Use this tool to visualize probabilities as volumes under a joint distribution.

2. Conditional Expectation Visualizer - Use this tool to visualize conditional expectations.

3. Convolution Visualizer - Use this tool to visualize convolutions.

Iterated Integrals and Sums¶

All definitions and results are available in Section 10.1.

1. The double integral of a surface $f(x,y)$ over a region $\mathcal{R}$ in the $x,y$ plane returns the volume, under the surface, over the region.

   • If $f(x,y)$ is equal to a constant, $c$, over $\mathcal{R}$ then the the volume equals $c \times \text{area}(\mathcal{R})$.

2. Double integrals can be computed using iterated integrals:

   $$\iint_{[x,y] \in \mathcal{R}} f(x,y) dx dy = \begin{cases} & \int_{x \in \mathcal{R}} \left(\int_{y \text{ such that } [x,y] \in \mathcal{R}} f(x,y) dy \right) dx \\ & \int_{y \in \mathcal{R}} \left(\int_{x \text{ such that } [x,y] \in \mathcal{R}} f(x,y) dx \right) dy \end{cases}$$

   (1)

   • Note that, when expanded as an iterated integral, the inner integral treats the outer variable as if it were constant.

   • The bounds of the inner integral may depend on the value of the outer variable.

3. Given a rectangular region, $\mathcal{R} = [a,b] \times [c,d]$:

   $$\iint_{[x,y] \in [a,b] \times [c,d]} f(x,y) dx dy = \begin{cases} & \int_{x =a}^b \left(\int_{y = c}^d f(x,y) dy \right) dx \\ & \int_{y =c}^d \left(\int_{x = a}^b f(x,y) dx \right) dy \end{cases}$$

   (2)

   • If $f(x,y) = g(x) \times h(y)$ then a double integral over a rectangle will factor into a product of univariate integrals:

   $$\iint_{[x,y] \in [a,b] \times [c,d]} f(x,y) dx dy = \left( \int_{x = a}^b g(x) dx \right) \times \left(\int_{y = c}^d h(y) dy \right)$$

   (3)

4. All of the same results hold for double sums

Conditional and Iterated Expectation¶

All definitions and results are available in Section 10.2.

1. The conditional expectation of $g(X,Y)$ over $Y$ given that $X = x$ is:

   $$\mathbb{E}_{Y|X = x}[g(x,Y)] = \mathbb{E}[g(X,Y)|X = x] = \begin{cases} \sum_{\text{all } y} g(x,y) \text{Pr}(Y = y|X = x) & \text{ if discrete} \\ \int_{\text{all } y} g(x,y) f_{Y|X = x}(y) & \text{ if continuous} \end{cases}$$

   (4)

   • The expression $\mathbb{E}_{\textbf{(a)}}[\textbf{(b)}]$ should be read “the expectation over (a) of (b).” A conditioning statement may appear in either the subscript, or inside the square brackets.

   • In general, the conditional expectation of one variable, given another, varies depending on the conditioning statement. For example, $\bar{y}(x) = \mathbb{E}[Y|X = x]$ is typically a function of $x$.

2. A joint expectation over two variables can be expressed as an iterated expectation:

   $$\mathbb{E}_{X,Y}[g(X,Y)] = \mathbb{E}_{X}[\mathbb{E}_{Y|X}[g(X,Y)]] = \mathbb{E}_{Y}[\mathbb{E}_{X|Y}[g(X,Y)]]$$

   (5)

   • An iterated expectation expands a joint expectation as a nested pair of expectations.

   • This rule expresses a joint average as an average of a conditional average.

   • In the case when $g(X,Y) = Y$, $\mathbb{E}[Y] = \mathbb{E}_X[\mathbb{E}[Y|X]]$, and when $g(X,Y) = X$, $\mathbb{E}[X] = \mathbb{E}_Y[\mathbb{E}[X|Y]].$

3. If $X$ and $Y$ are independent random variables then:

   $$\mathbb{E}_{X,Y}[g(X) \times h(Y)] = \mathbb{E}_{X}[g(X)] \times \mathbb{E}_{Y}[h(Y)]$$

   (6)

   • This rule does not apply when $X$ and $Y$ are dependent variables

Integration on Manifolds¶

All definitions and results are available in Section 10.3.

1. A manifold in the $x,y$ plane is a connected set of points, $[x,y]$ that form a smooth (continuous, differentiable) curve that does not cross itself.

   • The set of all $[x,y]$ such that $x + y = s$ is a manifold. It is equivalent to the line $y = s - x$.

   • The set of all $[x,y]$ such that $x^2 + y^2 = r^2$ is a manifold. It is equivalent to the circle with radius $r$.

2. The marginal mass/density function of the sum of two random variables is expressed as a sum/integral over a line. Given $S = X + Y$:

   $$\text{Pr}(S = s) = \sum_{\text{all }x} \text{Pr}(X = x,Y = s - x)$$

   (7)

   and

   $$f_{S}(s) = \int_{\text{all }x} f_{X,Y}(x,s - x) dx.$$

   (8)

   • If $X$ and $Y$ are independent, then the marginal mass/density function of $S$ is a convolution of the marginal mass/density functions of $X$ and $Y$:

   $$\text{Pr}(S = s) = \sum_{\text{all }x} \text{PMF}_X(x) \text{PMF}_Y(s - x) = \left[\text{PMF}_X * \text{PMF}_Y \right](s)$$

   (9)

   and

   $$f_{S}(s) = \int_{\text{all }x} f_{X}(x) f_Y(s - x) dx = [f_X * f_Y](s).$$

   (10)

3. To integrate over a circular region, we use an iterated integral in polar coordinates:

   $$\iint_{\text{all } x^2 + y^2 \leq r_*^2} f_{X,Y}(x,y) dx dy = \int_{r = 0}^{r_*} \left[\int_{\theta = 0}^{2 \pi} f_{X,Y}(r \cos(\theta), r \sin(\theta)) d\theta \right] r dr.$$

   (11)

   • The weighting by $r$ before $dr$ accounts for the fact that a circle, of radius $r$, has circumference proportional to $r$.

   • If $f_{X,Y}(x,y) = g(r)$ for some nonnegative function $g$, and $r = \sqrt{x^2 + y^2}$, then the density is rotationally symmetric. Then:

   $$\iint_{\text{all } x^2 + y^2 \leq r_*^2} f_{X,Y}(x,y) dx dy = 2 \pi \int_{r = 0}^{r_*} g(r) r dr.$$

   (12)