EXPECTED VALUE AND VARIANCE

Given a continuous random variable X and its PDF f_X(x), its expected value is defined as follows:

Quite often, we use the short-hand notation μ_X = E[X]. Again, this is straightforward extension of the discrete case, where E[X] ≡ ∑_i x_ip_X(x_i).

Example 7.2 As an illustration, let us consider the expected value of a uniform random variable on [a, b]. Symmetry suggests that the expected value should be the midpoint of the support. Indeed

By the same token, we define variance of a continuous random variable as:

Common shorthand notations for variance are σ² and ; its square root σ_X is standard deviation. More generally, we define the expected value of a function g(X) of a random variable as

The considerations we made about expected values of discrete random variables apply here as well. Since integration is a linear operator,⁴ just like the sum, expectation is linear in the continuous case, too. All of the properties of expectation and variance, that we have introduced for the discrete case, carry over to the continuous case. In particular, we recall the following very useful properties:

where α and β are arbitrary real numbers.