Question

Why does the linearity of expectation holds always even they are independent

Answer 1

Answer:

Explanation:

Linearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent.

The expected value of a random variable is essentially a weighted average of possible outcomes. We are often interested in the expected value of a sum of random variables. For example, suppose we are playing a game in which we take the sum of the numbers rolled on two six-sided dice:

Calculating the expected value of the sum of the rolls is tedious using our basic methods. Instead, we make the following argument: "Well, the expected value for each die is , and the two dice rolls are independent events, so the expected value for their sum should be ."

And this is true—these expected values add. But there’s more! The linearity property of expectation is especially powerful because it tells us that we can add expected values in this fashion even when the random variables are dependent.

Let that sink in for a moment, as it can be quite counter-intuitive! As an example, this means that the expected value for the amount of rain this weekend is precisely the expected value for the amount of rain on Saturday plus the expected value for the amount of rain on Sunday, even though we do not think that the amount of rain on Saturday is independent of the amount of rain on Sunday (for example, a very rainy Saturday increases the likelihood of a rainy Sunday).

On this page, we derive this property of expected value. We'll solve some basic problems, and then dive into the advanced techniques which allow us to solve many combinatorics problems, ranging from reasonably straight-forward to quite challenging. Finally, we’ll explore applications in other subject areas such as computer science and geometry.