Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.7 Original citation: Dougherty, C. (2012) EC220 - Introduction.

Christopher Dougherty

EC220 - Introduction to econometrics (review chapter)Slideshow: exercise r.7

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/141/

Available in LSE Learning Resources Online: May 2012

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EXERCISE R.7

R.7* Find E(X2) for X defined in Exercise R.2.

[R.2* A random variable X is defined to be the larger of the numbers when two dice are thrown, or the number if they are the same. Find the probability distribution for X.]

1

Definition of E[g(X)], the expected value of a function of X:

To find the expected value of a function of a random variable, you calculate all the possible values of the function, weight them by the corresponding probabilities, and sum the results.

n

iiinn pxgpxgpxgXgE

111 )()(...)()(

2

EXERCISE R.7

Definition of E[g(X)], 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 ...)(

3

EXERCISE R.7

n

iiinn pxgpxgpxgXgE

111 )()(...)()(

xi pi g(xi) g(xi ) pi

x1 p1 g(x1) g(x1) p1

x2 p2 g(x2) g(x2) p2

x3 p3 g(x3) g(x3) p3

… … …... ……... … … …... ……... xn pn g(xn) g(xn) pn

g(xi) pi

EXERCISE R.7

4

The calculation of the expected value of a function g(x) is shown in abstract in the table. The expected value is the sum of the terms g(xi)pi.

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

x1 p1 g(x1) g(x1) p1 1 1/36

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

x3 p3 g(x3) g(x3) p3 3 5/36

… … …... ……... 4 7/36

… … …... ……... 5 9/36

xn pn g(xn) g(xn) pn 6 11/36

g(xi) pi

EXERCISE R.7

In this exercise, X is the random variable defined in Exercise R.2. The 6 possible values of X and the corresponding probabilities are listed.

5

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

x1 p1 g(x1) g(x1) p1 1 1/36 1

x2 p2 g(x2) g(x2) p2 2 3/36 4

x3 p3 g(x3) g(x3) p3 3 5/36 9

… … …... ……... 4 7/36 16

… … …... ……... 5 9/36 25

xn pn g(xn) g(xn) pn 6 11/36 36

g(xi) pi

EXERCISE R.7

First you calculate the possible values of X2.

6

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

2 pi

x1 p1 g(x1) g(x1) p1 1 1/36 1 1/36

x2 p2 g(x2) g(x2) p2 2 3/36 4

x3 p3 g(x3) g(x3) p3 3 5/36 9

… … …... ……... 4 7/36 16

… … …... ……... 5 9/36 25

xn pn g(xn) g(xn) pn 6 11/36 36

g(xi) pi

EXERCISE R.7

The first value is 1 since x is equal to 1. The probability of X being equal to 1 is 1/36. Hence the weighted square is 1/36.

7

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

2 pi

x1 p1 g(x1) g(x1) p1 1 1/36 1 1/36

x2 p2 g(x2) g(x2) p2 2 3/36 4 12/36

x3 p3 g(x3) g(x3) p3 3 5/36 9 45/36

… … …... ……... 4 7/36 16 112/36

… … …... ……... 5 9/36 25 225/36

xn pn g(xn) g(xn) pn 6 11/36 36 396/36

g(xi) pi

EXERCISE R.7

Similarly for all the other possible values of X.

8

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

2 pi

x1 p1 g(x1) g(x1) p1 1 1/36 1 1/36

x2 p2 g(x2) g(x2) p2 2 3/36 4 12/36

x3 p3 g(x3) g(x3) p3 3 5/36 9 45/36

… … …... ……... 4 7/36 16 112/36

… … …... ……... 5 9/36 25 225/36

xn pn g(xn) g(xn) pn 6 11/36 36 396/36

g(xi) pi 791/36 = 21.97

EXERCISE R.7

The expected value of X2 is the sum of its weighted values in the final column. It is equal to 21.97. It is the average value of the figures in the previous column, taking the differing probabilities into account.

9

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

2 pi

x1 p1 g(x1) g(x1) p1 1 1/36 1 1/36

x2 p2 g(x2) g(x2) p2 2 3/36 4 12/36

x3 p3 g(x3) g(x3) p3 3 5/36 9 45/36

… … …... ……... 4 7/36 16 112/36

… … …... ……... 5 9/36 25 225/36

xn pn g(xn) g(xn) pn 6 11/36 36 396/36

g(xi) pi 791/36 = 21.97

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 4.47. Its square is 19.98.

EXERCISE R.7

10

E(X2) = 21.97 E(X) = 4.47[E(X)]2 = 19.98

Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use.

26.08.06