# Using the Uniform Distribution of Deaths assumption

It was my understanding that the Uniform Distribution of Deaths assumption (UDD) is that during a year people die at a constant rate, i.e.: for any two intervals within a year, if the intervals are the same length, then the same number of people will die during those intervals.

However, I am not getting questions on UDD correct, e.g.:

$$_.3q_{50.4} = 1 - _.3q_{50.4} = 1 - .3q_{50.4} \color{red}= 1 - .3q_{50}$$

The last equality uses my understanding of UDD and is what was marked wrong.

How is my understanding wrong? Surely if deaths are uniformly distributed, then the same number of people will die on $[50.4, 50.7]$ as on $[50, 50.3]$.

## 1 Answer

$q =$ number of deaths $/$ number of people, so it's a "death ratio". The deaths are uniformly distributed, but the death ratio is not. So:

$$_{.3}q_{50.4} \neq _{.3}q_{50}$$

## Example:

Let's say we start with $100$ people at year $50$ and end with $0$ at year $51$:

$$\mathcal{l}_{50}=100, \ \ \mathcal{l}_{51}=0$$

If deaths are distributed uniformly, we can linearly interpolate number of deaths:

$$_{.3}d_{x} = 30, \ \ 51 > x \geq 50$$

And we can linearly interpolate number of people alive:

$$\mathcal{l}_{50.4}=60$$

Hence,

\begin{align} _{.3}q_{50.4} =& _{.3}d_{50}/l_{50.4} = 30/60 \\ _{.3}q_{50} \ \ =& _{.3}d_{50}/l_{50} = 30/100 \end{align}