Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of

Definition of , the expected value of a function of X:

n

iiinn pxgpxgpxgXgE

111 ...

1

EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE

XgE

To find the expected value of a function of a random variable, one calculates all the possible values of the function, weights them by the corresponding probabilities, and sums the results.

Definition of , the expected value of a function of X:

Example:

For example, the expected value of X2 is found by calculating all its possible values, multiplying them by the corresponding probabilities, and summing.

n

iiinn pxpxpxXE

1

221

21

2 ...

n

iiinn pxgpxgpxgXgE

111 ...

2


XgE

3


xi pi g(xi) g(xi ) pi xi pi xi2 xi

2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

The calculation of the expected value of a function of a random variable will be outlined in general and then illustrated with an example.

4



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

First one makes a list of the possible values of X and the corresponding probabilities.

5



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

Next the function of X is calculated for each possible value of X.

6



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

Then, one at a time, the value of the function is weighted by its corresponding probability.

7



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

This is done individually for each possible value of X.

8



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

The sum of the weighted values is the expected value of the function of X.

9



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

The process will be illustrated for X2, where X is the random variable defined in the first sequence. The 11 possible values of X and the corresponding probabilities are listed.

10



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

First one calculates the possible values of X2.

11



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

The first value is 4, which arises when X is equal to 2. The probability of X being equal to 2 is 1/36, so the weighted function is 4/36, which we shall write in decimal form as 0.11.

12



2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

Similarly for all the other possible values of X.

13


The expected value of X2 is the sum of its weighted values in the final column. It is equal to 54.83. It is the average value of the figures in the previous column, taking the differing probabilities into account.


2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83


2 pi

x1 p1 g(x1) g(x1) p1 2 1/36 4 0.11

x2 p2 g(x2) g(x2) p2 3 2/36 9 0.50

x3 p3 g(x3) g(x3) p3 4 3/36 16 1.33

… … …... ……... 5 4/36 25 2.78

… … …... ……... 6 5/36 36 5.00

… … …... ……... 7 6/36 49 8.17

… … …... ……... 8 5/36 64 8.89

… … …... ……... 9 4/36 81 9.00

… … …... ……... 10 3/36 100 8.83

… … …... ……... 11 2/36 121 6.72

xn pn g(xn) g(xn) pn 12 1/36 144 4.00

g(xi) pi 54.83

Note that E(X2) is not the same thing as E(X), squared. In the previous sequence we saw that E(X) for this example was 7. Its square is 49.

14


7XE

83.542 XE

22 XEXE

Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.10.29

Documents

Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of