I have a question on my homework that I'm very befuddled about.
Find a value of a constant $c$ such that the formula $p(n)=\frac{c}{3^n}$ defines a probability distribution on the set $\{1,2,3,\ldots\}$ of all natural numbers. (Enter your answer as a decimal.)
I don't even know where to begin. All I can assume is that the probability should be equal for each real number. If there were $10$ numbers, each would have a $.1$ probability but the fact that this is infinite really confuses me especially since it has to satisfy this expression.
Firstly, you are not asked to define a probability on the reals, but rather on the naturals. Secondly, you seem to be under the impression that any two points should be assigned equal probability density, but that is not required at all. Note that $p(n)=\frac{c}{3^n}$ is already given, and that you are only asked to find the value of $c$ such that this will give a probability distribution. So, what property should a probability distribution on the naturals satisfy? In particular, $1=\sum ....$