Exploring Distribution Functions for the Time Until Death Variable

Ever thought about how an insurance company calculates the odds of you needing a payout? It might sound morbid, but this complex calculation is the bedrock of the entire life insurance industry. It’s all based on a concept actuaries call the future lifetime random variable, $T(x)$, which represents the time an individual aged $x$ will survive from this moment forward. To predict the likelihood of survival or death, we need the tools of probability, specifically, distribution functions. Let’s unpack these core concepts-the Cumulative Distribution Function (CDF), the Probability Density Function (PDF), and the crucial Force of Mortality-and see how they bring the math of life and death to life.

The cumulative distribution function: ${}_tq_x$ 🔗

Imagine you have a family member, let’s say your aunt, who is currently 60 years old. An insurance company needs to know the probability that she will pass away in the next 10 years. This exact probability is what the Cumulative Distribution Function (CDF) for the future lifetime variable $T(x)$ calculates. It’s denoted as ${}_tq_x$.

The notation ${}_tq_x$ reads as: “the probability that a life aged $x$ will die within the next $t$ years.” If your aunt is $x=60$ and $t=10$ years, then ${}_{10}q_{60}$ is the probability she dies between age 60 and 70. Actuarially, it’s a probability ranging from 0 (impossible) to 1 (certainty).

The complement of survival 🔗

The CDF, ${}_tq_x$, is intrinsically linked to its counterpart, the survival function, denoted as ${}_tp_x$. The term ${}_tp_x$ represents the probability that a life aged $x$ will survive for the next $t$ years. Since there are only two outcomes-survival or death-these two probabilities must add up to one:

$$ {}_tq_x + {}_tp_x = 1 $$

Therefore, you can think of the CDF, ${}_tq_x$, as the probability of “failure” (death) within the timeframe, which is the exact complement of the probability of “success” (survival). This relationship is fundamental to setting premium rates. For instance, The Insurance Regulatory and Development Authority of India (IRDAI) oversees how these probabilities inform the financial stability of insurance providers.

The probability density function: $f_{T(x)}(t)$ 🔗

While the CDF, ${}_tq_x$, gives us the probability of death within an *interval* of time (like 10 years), the Probability Density Function (PDF), $f_{T(x)}(t)$, gives us the *likelihood* of death occurring at a precise moment $t$ years from now. Since time until death is a continuous variable, we talk about density, not direct probability.

Conditional density of death 🔗

The PDF, $f_{T(x)}(t)$, represents the conditional density of death at time $t$, given that the individual has already survived to age $x$. It’s conditional because we start the clock at age $x$. The standard formula for this conditional PDF is:

$$ f_{T(x)}(t) = \frac{f(x+t)}{S(x)} $$

Here, $f(x+t)$ is the unconditional density of death at age $x+t$ (starting from birth), and $S(x)$ is the unconditional probability of surviving to age $x$ (also starting from birth). By dividing the density of death at age $x+t$ by the probability of having lived until $x$, we “condition” the probability on the fact that the person is currently alive at age $x$.

Think of it like a marathon. The unconditional density is the likelihood of someone finishing the race at the 42-kilometer mark, starting from the gun. The conditional density is the likelihood of them finishing at 42 kilometers, given that we know they have successfully made it to the 10-kilometer mark ($x$). The conditional probability focuses only on the remaining “future lifetime” of the event.

The force of mortality: $\mu(x+t)$ 🔗

The Force of Mortality, denoted $\mu(x)$, is arguably the most dynamic and informative of all the distribution concepts. It is the instantaneous rate of failure (or death) at an exact age $x$. It’s a key concept in actuarial science and demographic studies, including those conducted in the Indian context.

Understanding instantaneous hazard 🔗

To grasp $\mu(x)$, imagine an individual aged 80. The Force of Mortality $\mu(80)$ is the likelihood of that person dying right at that moment, $exactly$ at age 80, expressed as an annual rate. It’s a hazard rate, which for $T(x)$ is expressed as $\mu(x+t)$-the instantaneous rate of mortality at the attained age $x+t$.

Unlike the probability of death within a year, $q_x$ (which is an average probability over a time interval), the force of mortality is a rate that acts continuously. As a person ages, this rate generally increases because their risk of mortality rises. This is why mortality tables used by companies like Life Insurance Corporation of India (LIC) show a progressively steeper increase in the force of mortality for older ages.

Mathematically, the Force of Mortality $\mu(x)$ is defined as:

$$ \mu(x) = – \frac{1}{S(x)} \frac{d}{dx} S(x) $$

In simple terms, it’s the rate at which the survival function is dropping at age $x$, relative to the level of the survival function itself. A steep drop in the survival curve means a high force of mortality.

Linking PDF and the force of mortality 🔗

One of the most powerful insights in actuarial mathematics is the relationship that connects the instantaneous rate of failure (the Force of Mortality) to the likelihood of death at a specific time (the PDF). This relationship provides a clear, causal link:

$$ f_{T(x)}(t) = {}_tP_x \cdot \mu(x+t) $$

Survival meets instantaneous hazard 🔗

Let’s break down this crucial equation for $f_{T(x)}(t)$:

• ${}_tP_x$ (The Survival Component): This is the probability that the person survives the *entire interval* from age $x$ to age $x+t$. If the person doesn’t survive this time, the probability of them dying *at* age $x+t$ is zero.
• $\mu(x+t)$ (The Hazard Component): This is the instantaneous rate of death *at* the exact moment they reach age $x+t$.

So, the PDF is essentially saying: the density of death at time $t$ is the probability that you make it to that time ($\boldsymbol{{}_tP_x}$) multiplied by the instantaneous rate of death at that point ($\boldsymbol{\mu(x+t)}$). This relationship is fundamental for constructing consistent and accurate mortality models, which are constantly refined using real-world data, such as that collected by the Office of the Registrar General & Census Commissioner, India.

Applied distribution function problems 🔗

Actuarial problems often involve calculating specific probabilities using these distribution functions. A classic example is using a simple model like De Moivre’s Law to estimate these values. De Moivre’s Law assumes a uniform distribution of deaths, meaning the probability of dying is constant over each year of life, up to a maximum age (let’s call it $\omega$).

Example: De moivre’s law 🔗

If the maximum age ($\omega$) is 100, and a person is $x=60$:

• Survival Function ($\boldsymbol{{}_tp_x}$): The probability of surviving $t$ years is $\frac{\omega – (x+t)}{\omega – x}$. For $t=10$ years: $$ {}_{10}p_{60} = \frac{100 – 70}{100 – 60} = \frac{30}{40} = 0.75 $$ There is a 75% chance a 60-year-old survives to age 70.
• CDF ($\boldsymbol{{}_tq_x}$): The probability of death within $t$ years is $1 – {}_tp_x$. $$ {}_{10}q_{60} = 1 – 0.75 = 0.25 $$ There is a 25% chance of a 60-year-old dying before age 70.
• PDF ($\boldsymbol{f_{T(x)}(t)}$): The PDF is constant and equals the probability of death within the remaining lifetime divided by the remaining years of life: $$ f_{T(x)}(t) = \frac{1}{\omega – x} = \frac{1}{100 – 60} = \frac{1}{40} = 0.025 $$ This is the uniform density of death at any point between ages 60 and 100.

While simplistic, models like this help us understand how actuaries calculate probabilities of death within intervals-from a single year to a decade-by integrating the PDF over that time interval to get the CDF value, or by using the CDF’s complement to find the survival probability.

Understanding the CDF, PDF, and the Force of Mortality isn’t just an academic exercise. It’s the invisible mathematical framework that allows the entire global insurance industry to price risk, allocate capital, and ultimately, provide financial security to millions of families. It’s the science of making the unpredictable manageable.

What do you think? If the probability of surviving the next five years, ${}_5p_x$, is 0.98, what is the probability of dying within the next five years, ${}_5q_x$? How does the general increase in life expectancy in a country like India impact the calculations for the Force of Mortality ($\mu(x)$) for younger ages?

References

1. https://www.actuariesindia.org/subMenu.aspx?id=128
2. https://www.licindia.in/
3. https://censusindia.gov.in/census_and_npr/