1. Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables 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/http://learningresources.lse.ac.uk/141/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

2. 1 CONTINUOUS RANDOM VARIABLES A discrete random variable is one that can take only a finite set of values. The sum of the numbers when two dice are thrown is an example. 6 __ 36 5 __ 36 4 __ 36 3 __ 36 2 __ 36 2 __ 36 3 __ 36 5 __ 36 4 __ 36 probability 23 456789101112X 1 36 1 36

3. CONTINUOUS RANDOM VARIABLES Each value has associated with it a finite probability, which you can think of as a ‘packet’ of probability. The packets sum to unity because the variable must take one of the values. 6 __ 36 5 __ 36 4 __ 36 3 __ 36 2 __ 36 2 __ 36 3 __ 36 5 __ 36 4 __ 36 probability 23 456789101112X 1 36 1 36 2

4. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height However, most random variables encountered in econometrics are continuous. They can take any one of an infinite set of values defined over a range (or possibly, ranges). 3

5. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height As a simple example, take the temperature in a room. We will assume that it can be anywhere from 55 to 75 degrees Fahrenheit with equal probability within the range. 4

6. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height In the case of a continuous random variable, the probability of it being equal to a given finite value (for example, temperature equal to 55.473927) is always infinitesimal. 5

7. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height For this reason, you can only talk about the probability of a continuous random variable lying between two given values. The probability is represented graphically as an area. 6

8. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height 56 For example, you could measure the probability of the temperature being between 55 and 56, both measured exactly. 7

9. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES height 56 Given that the temperature lies anywhere between 55 and 75 with equal probability, the probability of it lying between 55 and 56 must be 0.05. 0.05 8

10. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES 57 height Similarly, the probability of the temperature lying between 56 and 57 is 0.05. 56 0.05 9

11. 55 607075 X 65 CONTINUOUS RANDOM VARIABLES 57 58 height And similarly for all the other one-degree intervals within the range. 0.05 10

12. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES 57 58 height The probability per unit interval is 0.05 and accordingly the area of the rectangle representing the probability of the temperature lying in any given unit interval is 0.05. 11

13. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES 57 58 height The probability per unit interval is called the probability density and it is equal to the height of the unit-interval rectangle. 12

14. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES 57 58 Mathematically, the probability density is written as a function of the variable, for example f(X). In this example, f(X) is 0.05 for 55 < X < 75 and it is zero elsewhere. height f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 13

15. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) 57 58 The vertical axis is given the label probability density, rather than height. f(X) is known as the probability density function and is shown graphically in the diagram as the thick black line. f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 14

16. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) Suppose that you wish to calculate the probability of the temperature lying between 65 and 70 degrees. f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 15

17. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) To do this, you should calculate the area under the probability density function between 65 and 70. f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 16

18. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) Typically you have to use the integral calculus to work out the area under a curve, but in this very simple example all you have to do is calculate the area of a rectangle. f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 17

19. 55 607075 X 65 0.05 CONTINUOUS RANDOM VARIABLES The height of the rectangle is 0.05 and its width is 5, so its area is 0.25. probability density f(X)f(X) 0.05 5 0.25 f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 18

20. 6575 X 70 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) Now suppose that the temperature can lie in the range 65 to 75 degrees, with uniformly decreasing probability as the temperature gets higher. 19

21. 6575 X 70 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) 0.20 0.15 0.10 0.05 The total area of the triangle is unity because the probability of the temperature lying in the 65 to 75 range is unity. Since the base of the triangle is 10, its height must be 0.20. 20

22. 6575 X 70 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) 0.20 0.15 0.10 0.05 In this example, the probability density function is a line of the form f(X) = b 1 + b 2 X. To pass through the points (65, 0.20) and (75, 0), b 1 must equal 1.50 and b 2 must equal -0.02. f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75 21

23. 6575 X 70 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) 0.20 0.15 0.10 0.05 Suppose that we are interested in finding the probability of the temperature lying between 65 and 70 degrees. f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75 22

24. 6575 X 70 CONTINUOUS RANDOM VARIABLES probability density f(X)f(X) 0.20 0.15 0.10 0.05 We could do this by evaluating the integral of the function over this range, but there is no need. f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75 23

25. 6575 X 70 CONTINUOUS RANDOM VARIABLES It is easy to show geometrically that the answer is 0.75. This completes the introduction to continuous random variables. probability density f(X)f(X) 0.20 0.15 0.10 0.05 f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75 24

26. Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.25