Hi! I've got a few problems that I wonder if someone could help me with.
1) If Y is a discrete random variable that assigns positive probabilites to only the positive integers, show that.
I don't have a clue about how to solve this one.
2) Let Y denote a geometric random variable with probability of success p.
a) Show that for a positive integer a,
b) Show that for positive integers a and b
I've been trying a lot with this one, but don't really know how to prove this.
3) Find E[Y(Y-1)] for a geometric random variable Y by finding d^2/dq^2 for
I don't know about this one either. My suggestion is that the second derivative is equal to this:
, but how do I use that to find E[Y(Y-1)]?
Would be nice if someone could help me with any of these problems.
The trick is simple, once you see it: express the sum as a double sum, then interchange the order of summation.
To see how to get from
to
take a piece of graph paper, label the axes j and k, and blacken the (j,k) points involved in the first sum. Then think about summing things in the other direction -- first k and then j.
I'm afraid, since this is my second post, that I don't know. :-) However, a good night's sleep has just cleared up the step: we're multiplying by an integer, which is equivalent to adding the muliplicand to itself the appropriate number of times. I'll have another stab at the next step before I ask for help.