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- Expected Value of a function g(R)
of a random Variable R that
has density function f(r) is defined as

- Two Common expectations associated with a random variable R
- The Mean (
) of R is
, which is the case g(R) = R.
That is, is a synonym for the expected value of a random variable. When there is more than one random variable defined, the name of the random variable is a subscript, e.g. ). Expectation of the random variable itself.

- The Variance
of R
is
.
( is a synonym for the variance of a random variable.) That is, define the function , then Var(R) = E[ g(R) ] The Standard Deviation is the square root of the variance. If the random variable is an estimator then its standard deviation is usually called its

**standard error**, denoted**se**.

Let, a,b, and c be non-random variables (constants) and let R and S be random variables- Properties of Means
- E[a] = a (expected value of a constant is itself)
- E[aR + b] = aE[R] + b (expectation is a linear operator)
- E[aR + bS + c] = aE[R] + bE[S] + c (E is linear no matter how many random variables are involved)
- If R and S are independent then E[RS] = E[R]E[S]
multiplying two random variables is not a linear operation ( R*R = R²). So only under special conditions does the expected value of the product equal the product of the expected values. THe special condition being independence, which is defined below.

- You can see the properties of expectation in action by running the tutorial for week 1.

Note that since is not a linear function of R, doing linear things to R does- Properties of Variance
**not**do linear things to Var(R).

- Var(a) = 0 (constants don't vary)
- Var(aR+b) = a²Var(R) (variance affected by square of scaling factor)
- Var(aR + bS ) = a²Var(R) + b²Var(S) + 2abCov(R,S) (Cov defined below)

- The Mean (
) of R is
, which is the case g(R) = R.

Document Last revised: 1997/1/5