The probability distribution of a random variable , like a distribution of data, has a mean and a standard deviation .
The law of large numbers says that the average of the values of observed in many trials must approach .
The mean is the balance point of the probability histogram or density curve. If is discrete with possible values having probabilities , the mean is the average of the values of , each weighted by its probability:
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The variance is the average squared deviation of the values of the variable from their mean. For a discrete random variable,
The standard deviation is the square root of the variance. The standard deviation measures the variability of the distribution about the mean. It is easiest to interpret for Normal distributions.
The mean and variance of a continuous random variable can be computed from the density curve, but to do so requires more advanced mathematics.
The means and variances of random variables obey the following rules. If and are fixed numbers, then
If and are any two random variables having correlation , then