How Do We Find Expected Values?
To find expected values of a discrete random variable, we follow a three-step process:
1.) Identify all the values the random variable in question can take
The resulting list of possible values may be infinitely long; e.g., $0,1,2,3,\dots $
2.) Find the probability of each of these possible values, and multiply the values and their probabilities
It may help to start with some examples before moving to the general case.
3.) Add the resulting expressions up.
This may require us to sum an infinite series; for instance, a geometric or exponential series, or some variation of these.