8.2 - Properties of Expectation

8.2 - Properties of Expectation

Example 8-6

Suppose the p.m.f. of the discrete random variable $$X$$ is:

 x 0 1 2 3 f(x) 0.2 0.1 0.4 0.3

What is $$E(2)$$? What is $$E(X)$$? And, what is $$E(2X)$$?

Theorem
When it exists, the mathematical expectation $$E$$ satisfies the following properties:
1. If $$c$$ is a constant, then $$E(c)=c$$
2. If $$c$$ is a constant and $$u$$ is a function, then:

$$E[cu(X)]=cE[u(X)]$$

Example 8-7

Let's return to the same discrete random variable $$X$$. That is, suppose the p.m.f. of the random variable $$X$$ is:

It can be easily shown that $$E(X^2)=4.4$$. What is $$E(2X+3X^2)$$?

Theorem

Let $$c_1$$ and $$c_2$$ be constants and $$u_1$$ and $$u_2$$ be functions. Then, when the mathematical expectation $$E$$ exists, it satisfies the following property:

$$E[c_1 u_1(X)+c_2 u_2(X)]=c_1E[u_1(X)]+c_2E[u_2(X)]$$

Before we look at the proof, it should be noted that the above property can be extended to more than two terms. That is:

$$E\left[\sum\limits_{i=1}^k c_i u_i(X)\right]=\sum\limits_{i=1}^k c_i E[u_i(X)]$$

Example 8-8

Suppose the p.m.f. of the discrete random variable $$X$$ is:

In the previous examples, we determined that $$E(X)=1.8$$ and $$E(X^2)=4.4$$. Knowing that, what is $$E(4X^2)$$ and $$E(3X+2X^2)$$?

Using part (b) of the first theorem, we can determine that:

$$E(4X^2)=4E(X^2)=4(4.4)=17.6$$

And using the second theorem, we can determine that:

$$E(3X+2X^2)=3E(X)+2E(X^2)=3(1.8)+2(4.4)=14.2$$

Example 8-9

Let $$u(X)=(X-c)^2$$ where $$c$$ is a constant. Suppose $$E[(X-c)^2]$$ exists. Find the value of $$c$$ that minimizes $$E[(X-c)^2]$$.

Note that the expectations $$E(X)$$ and $$E[(X-E(X))^2]$$ are so important that they deserve special attention.

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