Queens University at Kingston


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C. Expectations

Expected Value of a function g(R) of a random Variable R that has density function f(r) is defined as
$$E[R] \equiv \int_{-\infty}^{+\infty} g(r)r f(r) dr$$

Two Common expectations associated with a random variable R
  1. The Mean ($\mu$ ) of R is $\mu \equiv E[R]$ , which is the case g(R) = R.
    That is, $\mu$ 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. $\mu_R$ ). Expectation of the random variable itself.
  2. The Variance $\sigma^2$ of R is $Var[R] \equiv E[ (R-\mu)(R-\mu) ]$ . ($\sigma^2$ is a synonym for the variance of a random variable.) That is, define the function $g(R) = (R-m)^2$ , then Var(R) = E[ g(R) ]

    The Standard Deviation $\sigma$ 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.

Properties of Means

Let, a,b, and c be non-random variables (constants) and let R and S be random variables
  1. E[a] = a (expected value of a constant is itself)
  2. E[aR + b] = aE[R] + b (expectation is a linear operator)
  3. E[aR + bS + c] = aE[R] + bE[S] + c (E is linear no matter how many random variables are involved)
  4. 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.

Properties of Variance

Note that since $g(R) = (R-m)^2$ is not a linear function of R, doing linear things to R does not do linear things to Var(R).
  1. Var(a) = 0 (constants don't vary)
  2. Var(aR+b) = a²Var(R) (variance affected by square of scaling factor)
  3. Var(aR + bS ) = a²Var(R) + b²Var(S) + 2abCov(R,S) (Cov defined below)

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Document Last revised: 1997/1/5