Chapter 2

Hello

Could you be a little more specific as to which bits of the solution you're not sure about?

Thanks

Andy

 

Hello

1. Structure

Let's consider an example. Say we have a driver with 5 years of driving experience and we know whether or not the driver had an accident in each of those 5 prior years. Let's say the driver had an accident in years 2 and 5. According to the notation in the notes, this means we have:

\(y_1 = 0, y_2 = 1, y_3 = 0, y_4 = 0, y_5 = 1 \)

We can use this record to estimate the probability that the driver has an accident in year 6. One way to do this is to use the model in the notes, which takes this record and extracts two summary figures to use as inputs into a probability calculation. The summary figures used are:

• the total number of accidents that the driver has had; and
• the number of years for which we have a record of the driver.

In this case, our driver had 2 accidents in 5 years. Using the model in the notes we can estimate the probability as:

\( P[Y_{6} = 1 | y_1 = 0, y_2 = 1, y_3 = 0, y_4 = 0, y_5 = 1] = f(y_1 + y_2 + y_3 + y_4 + y_5) / g(5) = f(2) / g(5) \)

for some functions f and g. More generally, for a driver with m accidents in n years, we estimate the probability of them having an accident in year n+1 as \( f(m) / g(n) \) according to this model.

The model given in the notes doesn't specify what the functions f and g are specifically. All we're saying is:

• The model is based on two inputs - # of accidents and # of years of data
• This particular model uses a structure that splits these two inputs and applies separate functions to them, in this case using a ratio: \( f(m) / g(n) \)

This isn't the only way to do this, it is just one possible structure of a probability calculation that uses these inputs.

2. Why the functions are increasing

Let's consider another driver and compare them to our previous example. Let's say our second driver had 3 accidents (say years 1,2,3) in 5 years. So for this driver the probability of an accident in year 6 according to the model is:

\( P[Y_{6} = 1 | y_1 = 1, y_2 = 1, y_3 = 1, y_4 = 0, y_5 = 0] = f(3) / g(5) \)

Recall for our first driver (who had 2 accidents in 5 years) it was \( f(2) / g(5) \)

Now, the question is which of these do we think should be higher? It may make sense to assume that our second driver is more likely to have an accident in year 6 (as they had more accidents over the same period (5 years)). So, assuming this, we expect:

\( f(3) / g(5) > f(2) / g(5) \)

As the denominator is the same, we need \( f(3) > f(2) \). More generally, we expect f to be an increasing function.

Now consider a third driver with 2 accidents in 6 years (say in years 5 and 6). For this driver, the probability of an accident in year 7 according to the model is:

\( P[Y_{7} = 1 | y_1 = 0, y_2 = 0, y_3 = 0, y_4 = 0, y_5 = 1, y_6 = 1] = f(2) / g(6) \)

Let's again compare this to our first driver, which recall was \( f(2) / g(5) \). Would we expect this to be higher or lower compared to the expression for the third driver? Well they both had 2 accidents but our third driver had them over a longer period of time, so we may expect the probability of an accident in the next year for this driver to be lower than for driver 1, ie:

\( f(2) / g(6) < f(2) / g(5) \)

As the numerator is the same, we need \( g(6) > g(5) \). More generally, we expect g to be an increasing function.

Misc

Note this isn't the only possible model. For example, the model ignores when the accidents took place. Driver 2 had their accidents in the first three years but not years 4 and 5 - perhaps they are actually now a safer driver compared to driver 1 who had an accident last year?

Hope this helps

Andy