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 \(E(2X)\)?

This example leads us to a very helpful theorem.

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:

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

Proof

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:

formula

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

This example again leads us to a very helpful theorem.

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:

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

Proof

 

Example 8-8

Suppose the p.m.f. of the discrete random variable \(X\) is:

formula

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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