CT3-04 Page 29

[Simon C]

Hi John (or anyone else who wants to help!)

I am hoping you can clarify an item from CT3-04 Page 29.

In condition (ii), an item denoted o(h) is added for each of the three conditional probabilities under consideration. Unfortunately I am having trouble interpreting the meaning of o(h).

In particular I would have thought the three conditional probabilities displayed should sum to 1 as they would appear to be both exhaustive and mutually exclusive. However they sum to 1+3o(h).

I would be very grateful if you could help explain the meaning of o(h) in more detail and also let me know where my understanding of the conditional probabilities is going wrong.

Many thanks
Simon

 

[John Lee]

I would be very grateful if you could help explain the meaning of o(h) in more detail.

The o(h) is basically some other terms that will disappear when we differentiate (from 1st principles we will have f(x+h) - f(x) and will divide by h and let h tend to zero). so terms which have h² or higher powers in them will disappear (as when we have divided by h they'll still be a h left in them, whcih will head to zero.

For example, from page 2 of the tables:

exp(h) = 1 + h + h²/2! + h³/3! +...

We could write this:

exp(h) = 1 + h + o(h)

Similarly:

exp(-h) = 1- h + h²/2! - h³/3! +- ... = 1 - h + o(h)

In particular I would have thought the three conditional probabilities displayed should sum to 1 as they would appear to be both exhaustive and mutually exclusive...let me know where my understanding of the conditional probabilities is going wrong.

If we wrote out all of the terms of the summation then they would sum to 1. We just abreviated latter terms to o(h).

 

[Little Miss Actuary]

I think they are called landau symbols if you're interested. The o stands for order because you look at the order of the terms.

 

[Simon C]

Many thanks for the replies which were both very helpful. My understanding has improved though I’m still not quite 100% of the way there yet. To help with this, please could you confirm if my understanding below is correct:

- o(h) tends to zero as h tends to zero.

- For the three conditional probabilties in condition (ii), we are using o(h) to indicate that the other terms in the expressions for each of the three probabilities are of negligible value. This is true because h is very small and o(h) is therefore very small too.

- The three o(h) items in the three different conditional probabilities need not each be exactly the same collection of terms. This is how it is still possible for the three probabilities to sum to 1.

Thanks again.

 

[CA2 student]

- o(h) tends to zero as h tends to zero.

Yes

- For the three conditional probabilties in condition (ii), we are using o(h) to indicate that the other terms in the expressions for each of the three probabilities are of negligible value. This is true because h is very small and o(h) is therefore very small too.

Yes

- The three o(h) items in the three different conditional probabilities need not each be exactly the same collection of terms. This is how it is still possible for the three probabilities to sum to 1.

The three o(h) terms can not be the same collection of terms as they would all have the same value and would not sum to 0 (and hence the probability would not sum to 1). Think of the three o(h) terms as something like 0.00001, -0.000005 and -0.000005. Then they are all of negligible value and the three probabilities sum to 1.

 

[Simon C]

Thanks for that, all has become clear!