Edit: Actually though I think the answer can be gotten in the manner of this post. I think you'll just end up with
Show that E(X)=np when X is a binomial random variable.
We need to factor out np and change the limits to n-1
Now let y=x-1 so we get
Here's where I get stuck. Could anyone show me how to arrange this into a sum of the geometric series? All hints are welcome.
If is binomial, then the summation is wrong; your pmf has the kernal of a geometric random variable, but if is really geometric then you need to be summing over the positive integers.
I'll assume you actually want to find for geometric X; what you want to calculate is
In which case the gimmick is to notice that
The idea behind this is to interchange the infinite sum and the taking of the derivative, which is justified, though you would have to take a course in real analysis to see why.
If you really do want to calculate the expected value of a binomial, your pmf is totally wrong, and it would probably help if you used the right one, i.e. what undefined posted.
With this in mind, can you give me a hint on how to factor out np ( p is already factored out) and get the limits to y=0 to y=n-1? That is where I am having trouble. Thanks.