8.4 Chapter Summary

Interactive Tools:¶

1. Surface Visualizer - Use this tool to visualize surfaces.

Coordinate Systems¶

All definitions and results are available in Section 8.1.

1. A coordinate system is a rule for uniquely assigning positions in a $d$-dimensional space to $d$ values

   • A Cartesian coordinate system specifies locations with a list of $d$ numbers, which each measure the distance from the location, $x$, to the origin, along a series of perpendicular directions.

   • A polar coordinate system specifies locations in the plane with a distance from the origin, and an angle away from a reference direction (usually, the ray leaving the origin in the positive horizontal direction).

   • To convert between polar and Cartesian, use the rules:

   $$\text{Pol. to Cart. }\begin{cases} & x_1 = r \cos(\theta) \\ & x_1 = r \sin(\theta) \end{cases}, \quad \quad \text{Cart. to Pol. } \begin{cases} & r = \sqrt{x_1^2 + x_2^2} \\ & \theta = \tan^{-1}(x_2/x_1) \end{cases}$$

   (1)

2. A vector is an ordered list of finitely many numbers, $v = [v_1,v_2,...,v_d]$. The number of entries is the dimension of the vector.

   • Vectors represent positions in a Cartesian coordinate system. The zero vector is the vector $[0,0,...0]$. It points to the origin.

   • To add vectors, produce a new vector by adding their entries, $v + w = [v_1 + w_1, v_2 + w_2, ..., v_d + w_d]$.

   • To scale a vector, multiply it by a scalar, $\lambda v = [\lambda v_1, \lambda v_2, ..., \lambda v_d]$. Scaling a vector preserves its direction, but changes its length.

   • A linear combination of two vectors is a sum of the form $\alpha v + \beta w$ where $\alpha$ and $\beta$ are scalars, while $v$ and $w$ are vectors.

   • The length or magnitude of a vector is defined $\|v\| = \sqrt{v_1^2 + v_2^2 + ... v_d^2}$. A unit vector is a vector of length 1.

   • The direction of a vector may be uniquely represented with the matching unit vector $\hat{v} = v/\|v\|$. Two vectors, $v$ and $w$, are parallel if and only if $$\hat{v} = \pm \hat{w}$.

   • The angle, $\theta$, between two vectors may be measured using an inner (dot) product:

     $$v \cdot w = v_1 w_1 + v_2 w_2 + ... v_d w_d = \sum_{j=1}^d v_j w_j.$$

     (2)

     • Geometrically:

     $$v \cdot w = \|v\| \|w\| \cos(\theta).$$

     (3)

   • So, two vectors are perpendicualr (orthogonal) if and only if $v \cdot w = 0$.

Functions of Multiple Variables¶

All definitions and results are available in Section 8.2.

1. A function of multiple variables is a function that accpets multiple inputs (e.g. vectors).

   • A function is scalar-valued if it returns scalars. It is vector-valued if it returns vectors.

   • Scalar-valued functions of multiple variables define surfaces.

2. We can visualize surfaces using:

   • cross-sections: which are slices of the surface produced by fixing all but one input, then varying the free input,

   • level sets: which are all inputs that produce a fixed output value, or

   • heat maps: which encode the output value as a color.

Joint Distributions¶

All definitions and results are available in Section 8.3.

1. Discrete random vectors are characterized by joint probability mass functions which may be represented using a joint distribution table

   • Marginal mass functions are produced by either summing across the rows, or down the columns

   • Conditional mass functions are produced by isolating a row or column, and scaling by the corresponding marginal to normalize.

   • So, conditional mass functions are proportional to cross-sections of the table.

2. Continuous random vectors are characterized by joint density functions which are surfaces:

   • The joint density at $x = [x_1,x_2,...x_d]$ is the chance that every coordinate $X_j$ is within $\pm \frac{1}{2} \Delta x$ of $x_j$, scaled by the volume of the $d$-dimensional cube with side lengths $\Delta x$, in the limit as $\Delta x$ approaches zero.

   • The rules for computing chances, finding marginal densities, finding conditional densities, checking independence, and evaluating expectations are analogous to all of the same procedures we developed in Sections 1.5 and 1.6, as long as we work with densities instead of mass functions, and, replace all sums over possible values with integrals over possible vectors.