1.1 Mathematical expectation

Mathematical expectiation

MATHEMATICAL EXPECTIATION

If $f\left(x\right)$ is the p.d.f. of the random variable X of the discrete type with space R and if the summation

exists, then the sum is called the mathematical expectation or the expected value of the function $u\left(X\right)$ , and it is denoted by $E\left[u\left(x\right)\right]$ . That is,

We can think of the expected value $E\left[u\left(x\right)\right]$ as a weighted mean of $u(x)$ , $x\in R$ , where the weights are the probabilities $f\left(x\right)=P\left(X=x\right)$ .

There is another important observation that must be made about consistency of this definition. Certainly, this function $u\left(X\right)$ of the random variable X is itself a random variable, say Y . Suppose that we find the p.d.f. of Y to be $g\left(y\right)$ on the support $R_1$ . Then, $E\left(Y\right)$ is given by the summation ${\displaystyle \sum _{y\in R_1}yg\left(y\right)}$ .

In general it is true that ${\displaystyle \sum _{R}u\left(x\right)f\left(x\right)}={\displaystyle \sum _{y\in R_1}yg\left(y\right)}$ .

This is, the same expectation is obtained by either method.

When it exists, mathematical expectation E satisfies the following properties:

• If c is a constant, $E\left(c\right)=c,$
• If c is a constant and u is a function, $E\left[cu\left(X\right)\right]=cE\left[u\left(X\right)\right],$
• If $c_1$ and $c_2$ are constants and $u_1$ and $u_2$ are functions, then $E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]=c_{1}E\left[u_1\left(X\right)\right]+c_{2}E\left[u_2\left(X\right)\right].$

First, we have for the proof of (1) that $$E\left(c\right)={\displaystyle \sum _{R}cf\left(x\right)=c{\displaystyle \sum _{R}f\left(x\right)}}=c,$$ because ${\displaystyle \sum _{R}f\left(x\right)=1.}$

Next, to prove (2), we see that $$E\left[cu\left(X\right)\right]={\displaystyle \sum _{R}cu\left(x\right)f\left(x\right)=c{\displaystyle \sum _{R}u\left(x\right)f\left(x\right)=cE\left[u\left(X\right)\right]}}.$$

Finally, the proof of (3) is given by $$E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]={\displaystyle \sum _R\left[c_1{u}_1\left(x\right)+c_2{u}_2\left(x\right)\right]}f\left(x\right)={\displaystyle \sum _R{c}_1{u}_1\left(x\right)f\left(x\right)+{\displaystyle \sum _R{c}_2{u}_2\left(x\right)f\left(x\right)}}.$$

By applying (2), we obtain $$E\left[c_1{u}_1\left(X\right)+c_2{u}_2\left(X\right)\right]=c_{1}E\left[u_1\left(x\right)\right]+c_{2}E\left[u_2\left(x\right)\right].$$

Property (3) can be extended to more than two terms by mathematical induction; that is, we have (3') $$E\left[{\displaystyle \sum _{i=1}^k{c}_i{u}_i\left(X\right)}\right]={\displaystyle \sum _{i=1}^k{c}_{i}E\left[u_i\left(X\right)\right]}.$$

Because of property (3’), mathematical expectation E is called a linear or distributive operator .