May 2019 - Infinity is Really Big

May 30, 2019September 5, 2019

Systems of Linear Equations in Two Dimensions

Visual Linear Algebra Online, Section 1.3

Visualizing the parametric vector form of the solution set of a system with infinitely many solutions.

Jack and Jill earn a combined $79.50 when Jack works 2 hours and Jill works 3 hours. On the other hand, if Jack works 3 hours and Jill works 2 hours, they make $78.75 total. What are

… Read the rest

May 21, 2019September 5, 2019

Vectors in Two Dimensions

Visual Linear Algebra Online, Section 1.2

Image by Keith Johnston from Pixabay

A big-league baseball player smashes a four-seam fastball with the sweet-spot of the bat. Is it a home run? The answer is — it depends. It depends on the launch angle and on it being a “fair ball”. In other words, what is the

… Read the rest

May 17, 2019May 17, 2019

Linear and Trigonometric Force of Mortality

Studying for Exam LTAM, Part 1.11

Survival function for a continuous lifetime random random variable with a trigonometric (sinusoidal) force of mortality.

In the last post, “Transformed Power Function Survival Models”, we explored the consequences of assuming certain forms of the survival function for a continuous survival random variable $T_{x}$ .

Here we change our perspective. Instead of starting with … Read the rest

May 14, 2019June 12, 2020

Points, Coordinates, Skew Coordinates, and Graphs

Visual Linear Algebra Online, Chapter 1: Vectors, Matrices, and Linear Transformations (The First Chapter of an Online Textbook)

Section 1.1 (Referring to this Blog Post Titled “Points, Coordinates, Skew Coordinates, and Graphs” Above)

Determining the coordinates of two points in (slanted) skew coordinates.

What is the point? Wait a minute — that’s not the question I meant to

… Read the rest

May 7, 2019May 7, 2019

Transformed Power Function Survival Models

Studying for Exam LTAM, Part 1.10

A family of transformed power function survival functions $S_{0}(t)=1-\frac{t^{p}}{\omega^{p}}$ .

S_{0}(t)=1-\frac{t^{p}}{\omega^{p}} — A family of transformed power function survival functions $S_{0}(t)=1-\frac{t^{p}}{\omega^{p}}$ .

A power function is defined by a formula of the form $f(t)=at^{p}$ for some constants $a$ and $p$ .

For certain choices of $a$ and $p$ , such a function could serve as the force of mortality for a continuous survival random variable $T_{0}$ . But it could not serve as … Read the rest

May 1, 2019January 11, 2020

Curtate Expectation of Life

Studying for Exam LTAM, Part 1.9

Image by Clker-Free-Vector-Images from Pixabay

Means of random variables can be thought of as centers of mass. If the random variable is continuous, think of the region under the graph of its probability density function (PDF) as a thin sheet of metal with constant density. The mean will be at the center of mass

… Read the rest