I have a question about linearity if expectations related to the proof in the im

Question

I have a question about linearity if expectations related to the proof in the image below.
I understand that the expected value is the Sum of Xi*probability-i. I also know that this equals the summation of the expected[xi].
I do not understand what the Expectation of Xi is equal to *inside of a summation*.
That is, the expected value of a dice is {1+2...6}*1/6. When the expected value sign is brought on the inside of a summation, what is the expected value of xi? what is E[1]+E[2]...E[6]?
I suspect it relates to the double summation sign in the proof below but I do not understand with the double summation sign means.
Note: a previous answer to this quesrion used double integrals as an explanation. i need something much (much) more basic. assume im a novice. ideally will have worked examples
thank you! I have a question about linearity if expectations related to the proof in the image below.
I understand that the expected value is the Sum of Xi*probability-i. I also know that this equals the summation of the expected[xi].
I do not understand what the Expectation of Xi is equal to *inside of a summation*.
That is, the expected value of a dice is {1+2...6}*1/6. When the expected value sign is brought on the inside of a summation, what is the expected value of xi? what is E[1]+E[2]...E[6]?
I suspect it relates to the double summation sign in the proof below but I do not understand with the double summation sign means.
Note: a previous answer to this quesrion used double integrals as an explanation. i need something much (much) more basic. assume im a novice. ideally will have worked examples
thank you!
I understand that the expected value is the Sum of Xi*probability-i. I also know that this equals the summation of the expected[xi].
I do not understand what the Expectation of Xi is equal to *inside of a summation*.
That is, the expected value of a dice is {1+2...6}*1/6. When the expected value sign is brought on the inside of a summation, what is the expected value of xi? what is E[1]+E[2]...E[6]?
I suspect it relates to the double summation sign in the proof below but I do not understand with the double summation sign means.
Note: a previous answer to this quesrion used double integrals as an explanation. i need something much (much) more basic. assume im a novice. ideally will have worked examples
thank you! 17:02 AT&T; o ELX] (X1 (w) Xi(w)Pr [w] i 1 wen E [Xi], i 1 which was claimed. It can be shown that linearity of expectation also holds for countably infinite summa- tions in certain cases. For example, it holds that if 500, EllXill converges. 25% D

Explanation / Answer

Explanation

E(X) = sum{x.p(x)}. More elaborately, if X takes values values x1, x2, x3, ........, xn, then E(X) = sum{xi.p(xi)}.

Now, what will be E(X + Y) =?

Let us take an example. Suppose a pair of fair dice is rolled and S = sum of the numbers that roll up on the two dice. Then, S = 2, 3, 4, ........., 11, 12 with respective probabilities,

1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, 1/36, Hence, E(S) = 252/36 = 7.

But, S = sum of the number on Die1 + sum of the number on Die2. So, if X = number on Die1 and Y = number on Die2, then S = X + Y. Expected value for a single die is 21/6 = 3.5. Hence, E(X) = E(Y) = 3.5 and consequently, E(X) + E(Y) = 2 x 3.5 = 7 which is the same as E(S). Thus, E(X + Y)= E(X) + E(Y).

Or, in general, E(X1 + X2 + X3 + ..... + Xn) = E(X1) + E(X2) + E(X3) + ..... + E(Xn) or put in summation form, E{sum(Xi)} = Sum{E(Xi)}. DONE

In this context, it would be wothwhile to review some other properties of E.

1. E(constant) is the constant itself.

2.E(aX) = a.E(X), where 'a' is a constant.

3. E(a + bX) = a + b.E(X)

4. E{g(X)}} = sum{g(x).p(x)}. In particular, E(X2) = sum{x2.p(x)}

[and to guard against a popular mistake, E(X2) is not equal to

    

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