1. Review of Probability Concepts
ECON 4550
Econometrics
Memorial University of Newfoundland
Review of Probability Concepts
Appendix B
Adapted from Vera Tabakova’s notes

2. Appendix B: Review of Probability Concepts
B.1 Random Variables
B.2 Probability Distributions
B.3 Joint, Marginal and Conditional Probability Distributions
B.4 Properties of Probability Distributions
B.5 Some Important Probability Distributions
Principles of Econometrics, 3rd Edition

3. B.1 Random Variables
A random variable is a variable whose value is unknown until it is observed.
A discrete random variable can take only a limited, or countable, number of values.
A continuous random variable can take any value on an interval.
Principles of Econometrics, 3rd Edition

4. B.2 Probability Distributions
The probability of an event is its “limiting relative frequency,” or the proportion of time it occurs in the long-run.
The probability density function (pdf) for a discrete random variable indicates the probability of each possible value occurring.
Principles of Econometrics, 3rd Edition

5. B.2 Probability Distributions
Principles of Econometrics, 3rd Edition

6. B.2 Probability Distributions
Figure B.1 College Employment Probabilities
Principles of Econometrics, 3rd Edition

7. B.2 Probability Distributions
The cumulative distribution function (cdf) is an alternative way to represent probabilities. The cdf of the random variable X, denoted F(x), gives the probability that X is less than or equal to a specific value x
Principles of Econometrics, 3rd Edition

8. B.2 Probability Distributions
Principles of Econometrics, 3rd Edition

9. B.2 Probability Distributions
For example, a binomial random variable X is the number of successes in n independent trials of identical experiments with probability of success p.
(B.1)
Principles of Econometrics, 3rd Edition

10. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
Principles of Econometrics, 3rd Edition
Slide B-10 10

11. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
First: for winning only once in three weeks, likelihood is 0.189, see?
Times
Principles of Econometrics, 3rd Edition
Slide B-11 11

12. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks…
The likelihood of winning exactly 2 games, no more or less:
times
Principles of Econometrics, 3rd Edition
Slide B-12 12

13. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
So 3 times = is the likelihood of winning exactly 2 games
Principles of Econometrics, 3rd Edition
Slide B-13 13

14. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
And is the likelihood of winning exactly 3 games
Principles of Econometrics, 3rd Edition
Slide B-14 14

15. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
For winning only once in three weeks: likelihood is 0.189
0.441 is the likelihood of winning exactly 2 games
0.343 is the likelihood of winning exactly 3 games
So is how likely they are to win at least 2 games in the next 3 weeks
Principles of Econometrics, 3rd Edition
Slide B-15 15

16. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
So is how likely they are to win at least 2 games in the next 3 weeks
In STATA di binomial(3,1,0.7) di Binomial(n,k,p) is the likelihood of winning 1 or less
So we were looking for 1- binomial(3,1,0.7)
Principles of Econometrics, 3rd Edition
Slide B-16 16

17. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
So is how likely they are to win at least 2 games in the next 3 weeks
In SHAZAM, although there are similar commands, it is a bit more cumbersome
See for example:
Principles of Econometrics, 3rd Edition
Slide B-17 17

18. B.2 Probability Distributions
For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks
Try instead:
Principles of Econometrics, 3rd Edition
Slide B-18 18

19. B.2 Probability Distributions
Figure B.2 PDF of a continuous random variable
A continuous
variable
Principles of Econometrics, 3rd Edition

20. B.2 Probability Distributions
Principles of Econometrics, 3rd Edition

21. B.3 Joint, Marginal and Conditional Probability Distributions
Principles of Econometrics, 3rd Edition

22. B.3 Joint, Marginal and Conditional Probability Distributions
Principles of Econometrics, 3rd Edition

23. B.3.1 Marginal Distributions
Check error
In posted
slides!!!
Principles of Econometrics, 3rd Edition

24. B.3.1 Marginal Distributions
Principles of Econometrics, 3rd Edition

25. B.3.2 Conditional Probability
What is the likelihood
Of having an income
If the person has a
4-year degree? y
.04/.18=.22 1
.14/.18=.78
Principles of Econometrics, 3rd Edition

26. B.3.2 Conditional Probability
What is the likelihood
Of having an income
If the person has a
4-year degree?
78%
But only 69% in
general y
.04/.18=.22 1
.14/.18=.78
Principles of Econometrics, 3rd Edition
Slide B-26 26

27. B.3.2 Conditional Probability
Two random variables are statistically independent if the conditional probability that Y = y given that X = x, is the same as the unconditional probability that Y = y.
(B.4)
(B.5)
(B.6)
Principles of Econometrics, 3rd Edition

28. B.3.3 A Simple Experiment Y = 1 if shaded Y = 0 if clear
X = numerical value (1, 2, 3, or 4)
Principles of Econometrics, 3rd Edition

29. B.3.3 A Simple Experiment
Principles of Econometrics, 3rd Edition

30. B.3.3 A Simple Experiment
Principles of Econometrics, 3rd Edition

31. B.3.3 A Simple Experiment
Principles of Econometrics, 3rd Edition