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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 mX. • 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: m X E ( X ) x1 p1 x2 p2 xk pk k xi pi i 1 • 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. 2 s • The variance of a RV X is denoted by X. • The standard deviation of X is the square root of the variance, denoted by sX. 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: s x1 m X p1 x2 m X p2 xk m X pk 2 2 X k 2 x m i 1 2 i X 2 pi • 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 + bX, where X is a RV with mean mX and variance s X2 . • The mean of Y is: mY m abX a bm X • The variance of Y is: s s 2 Y 2 a bX b s 2 2 X Sums of Independent RVs • Let X and Y be independent random variables. Then m X Y m X m Y s 2 X Y 2 X s s 2 Y s 2 X Y s s 2 Y 2 X Sums of Dependent RVs • Let X and Y be dependent random variables. Then m X Y m X m Y s 2 X Y s s 2r s Xs Y s 2 X Y s s 2r s Xs Y 2 X 2 X 2 Y 2 Y 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 x approaches m.