1. Review of Probability Concepts
ECON 6002
Econometrics
Memorial University of Newfoundland
Review of Probability Concepts
Appendix B
SECOND
Adapted from Vera Tabakova’s notes

2. B.4 Properties of Probability Distributions
B.4.1 Mean, median and mode
For a discrete random variable the expected value is:
(B.7)
(B.8)
Where f is the discrete PDF of x
Principles of Econometrics, 3rd Edition

3. B.4.1 Mean, median and mode
For a continuous random variable the expected value is:
The mean has a flaw as a measure of the center of a probability distribution in that it can be pulled by extreme values.
Principles of Econometrics, 3rd Edition

4. B.4.1 Mean, median and mode
For a continuous distribution the median of X is the value m such that
In symmetric distributions, like the familiar “bell-shaped curve” of the normal distribution, the mean and median are equal.
The mode is the value of X at which the pdf is highest.
Principles of Econometrics, 3rd Edition

5. B.4.2 Expected values of functions of a random variable
Where g is any function of x, in particular;
(B.10)
Principles of Econometrics, 3rd Edition

6. B.4.2 Expected values of functions of a random variable
Principles of Econometrics, 3rd Edition

7. B.4.2 Expected values of functions of a random variable
The variance of a discrete or continuous random variable X is the expected value of
The variance
Principles of Econometrics, 3rd Edition
Slide B-7 7

8. B.4.2 Expected values of functions of a random variable
The variance of a random variable is important in characterizing the scale of measurement, and the spread of the probability distribution.
Algebraically, letting E(X) = μ,
(B.13)
Principles of Econometrics, 3rd Edition

9. B.4.2 Expected values of functions of a random variable
The variance of a constant is?
Principles of Econometrics, 3rd Edition
Slide B-9 9

10. B.4.2 Expected values of functions of a random variable
Figure B.3 Distributions with different variances
Principles of Econometrics, 3rd Edition

11. B.4.2 Expected values of functions of a random variable
Principles of Econometrics, 3rd Edition

12. B.4.2 Expected values of functions of a random variable
Principles of Econometrics, 3rd Edition

13. B.4.3 Expected values of several random variables
Principles of Econometrics, 3rd Edition

14. B.4.3 Expected values of several random variables
Principles of Econometrics, 3rd Edition

15. B.4.3 Expected values of several random variables
Figure B.4 Correlated data
Principles of Econometrics, 3rd Edition

16. Covariance and correlation coefficient
If X and Y are independent random variables then the covariance and correlation between them are zero. The converse of this relationship is not true.
(B.19)
(B.20)
Principles of Econometrics, 3rd Edition

17. Covariance and correlation coefficient
The correlation coefficient is a measure of linear correlation between the variables
Its values range from -1 (perfect negative correlation) and 1 (perfect positive correlation)
(B.20)
Principles of Econometrics, 3rd Edition
Slide B-17 17

18. B.4.3 Expected values of several random variables
If a and b are constants then:
(B.21)
(B.22)
(B.23)
Principles of Econometrics, 3rd Edition

19. B.4.3 Expected values of several random variables
If a and b are constants then:
(B.22)
So:
Why is that? (and of course the same happens for the case
of var(X-Y))
Principles of Econometrics, 3rd Edition
Slide B-19 19

20. B.4.3 Expected values of several random variables
If X and Y are independent then:
(B.24)
(B.25)
Principles of Econometrics, 3rd Edition

21. B.4.3 Expected values of several random variables
If X and Y are independent then:
Otherwise this expression would have to include all the doubling of each
of the (non-zero) pairwise covariances between variables
as summands as well
Principles of Econometrics, 3rd Edition
Slide B-21 21

22. B.4.4 The Simple Experiment Again
Principles of Econometrics, 3rd Edition

23. B.5 Some Important Probability Distributions
B.5.1 The Normal Distribution
If X is a normally distributed random variable with mean μ and variance σ2, it can be symbolized as
(B.26)
Principles of Econometrics, 3rd Edition

24. B.5.1 The Normal Distribution
Figure B.5a Normal Probability Density Functions with Means μ and Variance 1
Principles of Econometrics, 3rd Edition

25. B.5.1 The Normal Distribution
Figure B.5b Normal Probability Density Functions with Mean 0 and Variance σ2
Principles of Econometrics, 3rd Edition

26. B.5.1 The Normal Distribution
A standard normal random variable is one that has a normal probability density function with mean 0 and variance 1.
The cdf for the standardized normal variable Z is
(B.27)
Principles of Econometrics, 3rd Edition

27. B.5.1 The Normal Distribution
Principles of Econometrics, 3rd Edition

28. B.5.1 The Normal Distribution
A weighted sum of normal random variables has a normal distribution.
(B.27)
Principles of Econometrics, 3rd Edition

29. B.5.2 The Chi-square Distribution
Principles of Econometrics, 3rd Edition

30. B.5.2 The Chi-square Distribution
Figure B.6 The chi-square distribution
Principles of Econometrics, 3rd Edition

31. B.5.3 The t-Distribution
A “t” random variable (no upper case) is formed by dividing a standard normal random variable by the square root of an independent chi-square random variable, , that has been divided by its degrees of freedom m.
(B.34)
Principles of Econometrics, 3rd Edition

32. Figure B.7 The standard normal and t(3) probability density functions
B.5.3 The t-Distribution
Figure B.7 The standard normal and t(3) probability density functions
Principles of Econometrics, 3rd Edition

33. B.5.4 The F-Distribution
An F random variable is formed by the ratio of two independent chi-square random variables that have been divided by their degrees of freedom.
(B.35)
Principles of Econometrics, 3rd Edition

34. Figure B.8 The probability density function of an F random variable
B.5.4 The F-Distribution
Figure B.8 The probability density function of an F random variable
Principles of Econometrics, 3rd Edition

35. Keywords binary variable joint probability density function
binomial random variable
marginal distribution
cdf
mean
chi-square distribution
median
conditional pdf
mode
conditional probability
normal distribution
continuous random variable
pdf
correlation
probability
covariance
probability density function
cumulative distribution function
random variable
degrees of freedom
standard deviation
discrete random variable
standard normal distribution
expected value
statistical independence
experiment
variance
F-distribution
Principles of Econometrics, 3rd Edition