## SECTION 4.5 Summary

• 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:

236

• 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

If and are independent, then . In this case,