Simple question that I've been staring at for too long to think about sensibly.
Given , prove that .
Now obviously can easily prove using formula for binomial distribution that , and on up the sequence to . But I don't know how to prove in general that the sums of the two series are such that .
If I write the two sequences as sums I get,
Am I on the right track...?
I understand that .
I also get that the sum of the two sequences must equal 1 because, due to their respective distributions, the two events, and partition the sample space, so that .
Since is the probability of success and is the probability of failure of the same experiment, the probability that there are successes in Bernouli trials must equal the probability that there are failures in the same Bernouli trials.
Probably I'm being a bit dense, but I don't understanding how to prove it.
[I am a bit confused by the notation as well. If I change the notation per your suggestion, then aren't I expressing the sum ? Does it make sense to write that?]
You do want do you not
Now there will be some overlapping there.
You know that
This is a post script.
You have two distinct cases.
If the proof is straight forward.
In the first sum when and the second sum then the two corresponding terms are equal.
However, if you must take special care.
You have overlapping terms is each sum.