probability density function Archives - Infinity is Really Big

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 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

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 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

April 1, 2019April 23, 2019

Families of Continuous Survival Random Variables

Studying for Exam LTAM, Part 1.1

Where do we begin? Assuming the reader’s knowledge of the fundamentals of (calculus-based) probability theory, the best place to start is with continuous survival models. This is also a common initial chapter in the textbooks on the subject.

As I mentioned in the post “Studying for Exam … Read the rest