1. 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

2. 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. 2 CONTINUOUS RANDOM VARIABLES 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. 55 607075 X 65 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 CONTINUOUS RANDOM VARIABLES

4. 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 CONTINUOUS RANDOM VARIABLES 55 607075 X 65 height

5. 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 CONTINUOUS RANDOM VARIABLES 55 607075 X 65 height

6. 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 CONTINUOUS RANDOM VARIABLES 55 607075 X 65 height

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

8. 8 55 607075 X 65 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 CONTINUOUS RANDOM VARIABLES

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

10. 10 55 607075 X 65 height 56 0.05 5758 And similarly for all the other one-degree intervals within the range. CONTINUOUS RANDOM VARIABLES

11. 11 55 607075 X 65 height 565758 0.05 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. CONTINUOUS RANDOM VARIABLES

12. 12 55 607075 X 65 height 565758 0.05 The probability per unit interval is called the probability density and it is equal to the height of the unit-interval rectangle. CONTINUOUS RANDOM VARIABLES

13. 13 55 607075 X 65 height 565758 0.05 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. f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75 CONTINUOUS RANDOM VARIABLES

14. 14 55 607075 X 65565758 0.05 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. probability density f(X)f(X) CONTINUOUS RANDOM VARIABLES f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75

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

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

17. 17 55 607075 X 65 0.05 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. CONTINUOUS RANDOM VARIABLES f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75

18. 18 55 607075 X 6556 0.05 probability density f(X)f(X) The height of the rectangle is 0.05 and its width is 5, so its area is 0.25. 0.25 CONTINUOUS RANDOM VARIABLES f(X) = 0.05 for 55 X 75 f(X) = 0 for X 75

19. 19 7075 X 65 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. CONTINUOUS RANDOM VARIABLES

20. 20 7075 X 65 probability density f(X)f(X) 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. 0.20 0.15 0.10 0.05 CONTINUOUS RANDOM VARIABLES

21. 21 7075 X 65 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 CONTINUOUS RANDOM VARIABLES

22. 22 7075 X 65 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. CONTINUOUS RANDOM VARIABLES f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75

23. 23 7075 X 65 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. CONTINUOUS RANDOM VARIABLES f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75

24. 24 7075 X 65 probability density f(X)f(X) 0.20 0.15 0.10 0.05 It is easy to show geometrically that the answer is 0.75. This completes the introduction to continuous random variables. CONTINUOUS RANDOM VARIABLES f(X) = 1.50 – 0.02X for 65 X 75 f(X) = 0 for X 75

25. Copyright Christopher Dougherty 2012. 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 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 EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.10.29