Let be a sequence of events of a sample space . Find a sequence of mutually exclusive events such that for all , .

- Sep 8th 2010, 10:19 AMZennieProbability question involving sequences of mutually exclusive events.
Let be a sequence of events of a sample space . Find a sequence of mutually exclusive events such that for all , .

- Sep 8th 2010, 10:29 PMFailure
- Sep 8th 2010, 11:34 PMZennie
Thanks, Failure. You wouldn't be able to explain how you got your answer would you? I don't exactly understand the question... which means I don't really understand your answer either.

- Sep 9th 2010, 12:19 AMFailure
You want a sequence of mutually exclusive events (i.e. disjoint sets) that has the same union as the sequence .

So let's start by setting . Now for we can take but in order to keep it disjoint from we take away . - So and are disjoint, agreed?

Similarly, for we can take , but in order to keep it disjoint from must take away their union, or, what amounts to the same thing . We choose .

So generally, we would like to define .

Observe that , for all i, by definition. Also, if we see that

Finally, to show that .

First, from for all i it follows immediately that

Second, suppose that . In that case there must exist a smallest , such that .

But then we have that and therefore , a contradiction. Thus there can be no such x, hence we have that