The Mean of a Discrete RV The mean

The Mean of a Discrete RV • The mean of a RV is the average value the RV takes over the long-run. – The mean of a RV is analogous to the mean of a large population. – The mean of a RV is different than a sample mean, which is the average of a sample of size n taken from a population. • The mean of the RV X is denoted by m. X. • The mean is also called the expected value, denoted E(X).

The Mean of a Discrete RV • The mean of a discrete RV X that takes k different values with probability pi for the ith value, the mean is: • The mean is the sum of the values of the RV, weighted by the probabilities of the values.

The Variance of a Discrete RV • The variance of a RV is a measure of the spread in the probability distribution of the RV about the mean. – The variance of a RV is analogous to the variance of a large population. – The variance of a RV is different than the sample variance. • The variance of a RV X is denoted by. • The standard deviation of X is the square root of the variance, denoted by s. X.

The Variance of a Discrete RV • For a discrete RV X that takes k different values with probability pi for the ith value, the variance is: • The variance is a sum of the squared distances between the values of the RV and its mean, weighted by the probabilities of the values.

Mean & Variance of Continuous RVs • We can find the mean and variance of a continuous random variable, but we need to use calculus techniques to do so. • Beyond the scope of MATH 106.

Mean & Variance of a Linear Function of a RV • Let Y = a + b. X, where X is a RV with mean m. X and variance. • The mean of Y is: • The variance of Y is:

Sums of Independent RVs • Let X and Y be independent random variables. Then

Sums of Dependent RVs • Let X and Y be dependent random variables. Then where r is the correlation between the random variables X and Y.

The Law of Large Numbers • As the sample size n (from a population with finite mean m) increases without bound, the sample mean approaches m.