Why do we need to find E(X^2) separately? Why can\'t we just split it into E(X^2

Question

Why do we need to find E(X^2) separately? Why can't we just split it into E(X^2) = E(X * X) = E(X) * E(X) = 1/2 * 1/2?

3. Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on the interval [0, 1]. Determine the value of 3. +2E(XX3) 4E(X2X3) Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Section 4.2. Properties of Expectations Since X1, X2, and X are independent, E(XiXj) = E(X)E(Xj) for i j. Therefore, the above expectation can be written in the form: Also, since each has the uniform distribution on the interval [0,1], then E(Xi)- and 0 Hence, the desired expectation has the value 1/2.

Explanation / Answer

No, we cannot go according to that procedure because for any two dependent event E(X^2) is not equal to (E(X))^2.

Since two events are dependent, they are likely to influence the expected value of each of the event. Hence treating these separately could not convey the right expected value .

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