Where x is any event, . The probability of an event ranges between impossible and certain. Where X is the set of all possible events x, . The probability that some possible event or other occurs is certain. If random variables are independent, then their covariance and correlation is zero.
Covariance ranges over the real numbers.
If is high, then it is possible to make good predictions about one variable if you know the other. If you observe all members of a population, then it is straightforward to calculate the mean and the variance. However, in many cases we observe only PART of the population - and then use what we have seen to estimate what the whole population is like. Think about the variance of a single point. Notice that we could have just used ONE observation instead of the sample mean.