Standard actuarial notation for future lifetime distributions

Welcome to the Language of Life and Death!

In Exam FAM, you’ll encounter a lot of "shorthand" notation. Think of actuarial notation as a secret code that allows us to write long, complex sentences about probabilities using just a few letters and numbers. Instead of writing "The probability that a person currently aged 40 survives for at least another 10 years," we just write \({}_{10}p_{40}\).

Don't worry if this seems like alphabet soup at first. Once you understand the pattern, you'll be reading these symbols like a pro. This chapter is the foundation for almost everything else in the Mortality Models section, so let's get the basics right!

1. The Core Variables: \(T_x\) and \(K_x\)

Before we look at probabilities, we need to define the random variables that represent how much time is left.

\(T_x\) (Future Lifetime): This is a continuous random variable representing the exact amount of time (down to the second) from age \(x\) until the moment of death.
\(K_x\) (Curtate Future Lifetime): This is a discrete random variable representing the number of completed years lived before death. It’s like counting how many more birthdays someone will celebrate.

Quick Analogy:
Imagine you are running a race. \(T_x\) is the exact time on the stopwatch when you cross the finish line. \(K_x\) is the number of full laps you completed before the race ended.

Key Takeaway: \(T_x\) is precise (continuous), while \(K_x\) is the "integer part" of \(T_x\) (discrete).

2. The "Big Two": Survive or Die (\(p\) and \(q\))

The most common symbols you will see are \(p\) and \(q\). They represent the two possible outcomes: you either stay alive or you don't.

\({}_t p_x\): Survival Probability

This is the probability that a life aged \(x\) survives for at least \(t\) more years.
Formula: \({}_t p_x = P(T_x > t)\)

\({}_t q_x\): Mortality (Death) Probability

This is the probability that a life aged \(x\) dies within the next \(t\) years.
Formula: \({}_t q_x = P(T_x \le t)\)

How to Read the Subscripts:

• The right subscript (\(x\)) is the current age of the person.
• The left subscript (\(t\)) is the duration of time we are looking at.
• Important Rule: If the duration \(t\) is 1 year, we usually leave it out. So, \(p_x\) is the same as \({}_1 p_x\).

Memory Aid:
p stands for Pass (you passed the test of time and survived).
q stands for Quit (you "quit" life during that period).

Quick Review Box:
Since you must either survive or die, these two probabilities always add up to 1:
\({}_t p_x + {}_t q_x = 1\)

3. Deferred Mortality: \({}_{u|t} q_x\)

Sometimes we want to know the probability that someone survives for a while, but then dies during a specific later period. This is called deferred mortality.

The Symbol: \({}_{u|t} q_x\)
The Meaning: The probability that a life aged \(x\) survives for \(u\) years, but then dies within the \(t\) years after that.

Step-by-Step Breakdown:
Think of this in two steps:
1. The person must survive the first \(u\) years (the "waiting period").
2. Then, they must die in the next \(t\) years (the "event window").

Common Formulas:
1. \({}_{u|t} q_x = {}_u p_x - {}_{u+t} p_x\) (Surviving \(u\) years minus surviving \(u+t\) years).
2. \({}_{u|t} q_x = {}_u p_x \cdot {}_t q_{x+u}\) (Survival to age \(x+u\) times the probability of dying in the next \(t\) years).

Common Mistake to Avoid:
Students often forget the age shift in the second formula. Remember, after surviving \(u\) years, the person is now aged \(x+u\), so you must use \(q_{x+u}\), not \(q_x\)!

4. The Force of Mortality (\(\mu_{x+t}\))

The Force of Mortality is the "instantaneous" rate of death. It’s like looking at the speedometer of a car to see how fast you are going at one specific second, rather than your average speed for the whole trip.

The Concept: It represents the likelihood of death at an exact moment, given that the individual has survived until that moment.

The Relationship to Survival:
The most important formula to memorize for the exam is the relationship between survival and the force of mortality:
\({}_t p_x = \exp\left( -\int_0^t \mu_{x+s} ds \right)\)

Did you know? If the force of mortality is constant (let's call it \(\mu\)), then the survival function simplifies to a very easy formula: \({}_t p_x = e^{-\mu t}\). This is just like the exponential distribution you might have seen in probability class!

5. Expectation of Life

In simple terms, this is the "average" number of years someone has left to live.

\(\mathring{e}_x\) (Complete Expectation of Life)

This uses the continuous variable \(T_x\). It is the average total future lifetime.
Formula: \(\mathring{e}_x = \int_0^{\infty} {}_t p_x dt\)

\(e_x\) (Curtate Expectation of Life)

This uses the discrete variable \(K_x\). It counts only the average number of full years survived.
Formula: \(e_x = \sum_{k=1}^{\infty} {}_k p_x\)

Memory Tip:
The "ring" or "circle" above the \(\mathring{e}\) looks like an "O" for "Ongoing" or "Overall" time (continuous). The plain \(e\) is for "Easy" counting (discrete years).

Summary Checklist

Before moving to the next chapter, make sure you can answer these:
1. Do I know the difference between \(p\) (survive) and \(q\) (die)?
2. Can I identify the current age (\(x\)) and the duration (\(t\)) in a symbol?
3. Do I understand that \({}_t p_x + {}_t q_x = 1\)?
4. Can I explain deferred mortality as a "wait then die" probability?
5. Do I recognize \(\mu\) as the instantaneous rate of death?

Don't worry if this feels like a lot of notation! As you practice more problems, these symbols will become second nature, like reading a second language. You've got this!