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.

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Presentation transcript:

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  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 p i for the i th 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  X.

The Variance of a Discrete RV For a discrete RV X that takes k different values with probability p i for the i th 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 + bX, where X is a RV with mean  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  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  ) increases without bound, the sample mean approaches .