20.3 Properties of the expected value

As usual, the expected value has several useful properties. Most importantly, the expected value is linear with respect to the random variable.

Theorem 129. (Linearity of the expected value)

Let (Ω,Σ,P) be a probability space, and let X,Y : Ω → ℝ be two random variables. Moreover, let a,b ∈ℝ be two scalars. Then

holds.

We are not going to prove this theorem here, but know that linearity is an essential tool. Do you recall the game that we used to introduce the expected value for discrete random variables? I toss a coin, and if it comes up heads, you win $1. Tails, you lose $2. If you think about it for a minute, this is the

distribution, and as such,

Of course, linearity goes way beyond this simple example. As you’ve gotten used to this already, linearity is a crucial property in mathematics. We love linearity.

Remark 20.

Notice that Theorem 129 did not say that X and Y have...