Mean and Variance
The mean, or first moment, of a distribution is a measure of the average. Suppose that a random variable has three outcomes.
To calculate the mean of X, we compute E(X). That is,E(X) = X = .2(3) + .6(5) + .2(12) = 6.0
The variance of X is calculated as E(X - X)2. We can augment our table as follows:
Now, we take E(X - X)2.E(X - X)2 = .2(9) + .6(1) + .2(36) = 9.6
Suppose that the values of X were raised to 4, 6, and 13. What do you think would happen to the mean of X? What do you think would happen to the variance of X? Verify your guesses by setting up the table and doing the calculation.
One way to understand the relationship between E(X2) and the variance of X is to write out the following identity.
The standard deviation of a random variable is the square root of the variance. In the example above, the standard deviation would be the square root of (9.6).
The mean of X is written as mX The Greek letter is pronounced "mew," although it often is transliterated as "mu." The standard deviation of X is written as sX. The Greek letter is called "sigma." Using Greek notation, the variance is written as s2X
Often, we will take two random variables, X and Y, and add them to create a new random variable. We could give the new random variable its own name, Z, but often we just call it X+Y.
The properties of the expectation operator imply that:
The term sXY is called the covariance of X and Y. We will return to it later in the course. For now, we note that in the case where X and Y are independent, the covariance is 0, and the equation reduces to:
It follows that if we have n independent random variables X that have the same mean mX and variance s2X, and we call the sum of these random variables V, then
These are called iid equations, because they refer to the sum of indepent, identically distributed random variables. Verify that the iid equations are correct.