Infinity is Really Big

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 $S_{0}(t)=1-\frac{t^{p}}{\omega^{p}}$ .

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

April 30, 2019May 8, 2019

Curtate Future Lifetime Random Variable

Studying for Exam LTAM, Part 1.8

When somebody asks you how old you are, how do you answer? Are you like most people? Do you give your age from your previous birthday? Or do you get extra-precise and say something like “I am 21 years, 5 months, and 1 week old”?

If you answer … Read the rest

April 23, 2019April 30, 2019

Gompertz-Makeham Survival Model

Studying for Exam LTAM, Part 1.7

You susceptibility to death generally grows higher as you age. This is an unfortunate fact of life. However, life insurance can help protect your family if you should die. It is a good idea to purchase life insurance, especially if you have dependents.

So far we have explored … Read the rest

April 18, 2019April 23, 2019

Measures of Spread in Survival Models

Studying for Exam LTAM, Part 1.6

One of the main lessons students should get out of any statistics course is the fact that quantitative data can be described with summary measures.

Data come with a wide variety of “locations”, “spreads”, and “shapes”. These words refer to the nature of the distribution of a … Read the rest

April 10, 2019April 23, 2019

Triangular Survival Models

Studying for Exam LTAM, Part 1.5

*An example of a triangular distribution probability density function (PDF).*

An important principle in mathematical modeling is to start simple. For example, if you have a situation where a linear function matches the trend in your data, go ahead and use a linear function rather than, for example, a quadratic function.

If a … Read the rest

April 9, 2019April 23, 2019

Constant Force of Mortality (Exponential Distribution)

Studying for Exam LTAM, Part 1.4

Survival random variables apply to other situations besides the lifetimes of people. A common example in textbooks is the lifetime of a light bulb.

Modern-day light bulb technology seems to be much better than it was even thirty years ago. I think this is especially true with regard … Read the rest

April 4, 2019April 23, 2019

The Complete Expectation of Life

Studying for Exam LTAM, Part 1.3

What is your life expectancy? Are you a smoker? A heavy drinker? Male or Female? And how well do you eat and exercise?

These factors can all affect how long you can “expect” to live.

However, the word “expect” can be misleading in this context. It is an … Read the rest

April 2, 2019April 23, 2019

The Force of Mortality (Hazard Rate Function)

Studying for Exam LTAM, Part 1.2

We all face potential risks to our lives every day. Daily risks include traffic accidents, lightening strikes, and poisoning. These risks can be quantified mathematically in numerous ways.

Fortunately, succumbing to a life-endangering risk on any given day has a low probability of occurrence. However, if you have … Read the rest