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The Mean of a Discrete RV

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.

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The Mean of a Discrete RV

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  1. 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).

  2. 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.

  3. 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 sX.

  4. 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.

  5. 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.

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

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

  8. 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.

  9. 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.

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