ECON 4550
Econometrics
Memorial University of Newfoundland
Review of Probability Concepts
Adapted from Vera Tabakova’s notes

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
Slide B-2

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
Slide B-3

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.
f ( x)  P  X  x 
f ( x1 )  f ( x2 ) 
Principles of Econometrics, 3rd Edition
 f ( xn )  1
Slide B-4
Principles of Econometrics, 3rd Edition
Slide B-5
Figure B.1 College Employment Probabilities
Principles of Econometrics, 3rd Edition
Slide B-6

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
F  x  P  X  x
Principles of Econometrics, 3rd Edition
Slide B-7
Principles of Econometrics, 3rd Edition
Slide B-8

For example, a binomial random variable X is the number of
successes in n independent trials of identical experiments with
probability of success p.
n x
P  X  x   f  x     p (1  p ) n x
 x
 n
n!
where n!  n  (n  1 n  2 
 
 x  x! n  x !
Principles of Econometrics, 3rd Edition
(B.1)
 21
Slide B-9

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

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?
n!
x!
nx
!


3!
1!
31
!
3
Times
px 
1 pnx 0. 7 1 
1 0. 72 0. 063
Principles of Econometrics, 3rd Edition
Slide B-11

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:
n!
3!

 3!  321 3
211
x!
nx
!
2!
32
!
2!
1
!
times
px 
1 pnx
0. 7 2 
1 0. 71 0. 147
Principles of Econometrics, 3rd Edition
Slide B-12

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 0.147 = 0.441 is the likelihood of winning exactly 2 games
Principles of Econometrics, 3rd Edition
Slide B-13

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 0.343 is the likelihood of winning exactly 3 games
n!
x!
nx
!

3!
3!
33
!

3!
3!1

3!
3!1
1
px 
1 pnx 0. 7 3 
1 0. 70 0. 343
Principles of Econometrics, 3rd Edition
Slide B-14

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 0.784 is how likely they are to win at least 2 games in the next 3
weeks
Principles of Econometrics, 3rd Edition
Slide B-15

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 0.784 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

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 0.784 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:
http://shazam.econ.ubc.ca/intro/stat3.htm
Principles of Econometrics, 3rd Edition
Slide B-17

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:

http://www.zweigmedia.com/ThirdEdSite/stats/be
rnoulli.html
Principles of Econometrics, 3rd Edition
Slide B-18
A continuous
variable
Figure B.2 PDF of a continuous random variable
Principles of Econometrics, 3rd Edition
Slide B-19
P  20  X  40  
40
 f  x  dx .355
20
P  X  x 
x
 f  t  dt  F  x 

P  20  X  40   F (40)  F (20)  .649  .294  .355
Principles of Econometrics, 3rd Edition
Slide B-20
1
2

X 
3
4
high school diploma or less
some college
four year college degree
advanced degree
0 if had no money earnings in 2002
Y 
1 if had positive money earnings in 2002
Principles of Econometrics, 3rd Edition
Slide B-21
 f  x, y   1
x
Principles of Econometrics, 3rd Edition
y
Slide B-22
f X ( x )   f ( x, y )
for each value X can take
y
fY ( y )   f ( x, y )
(B.2)
for each value Y can take
x
Check error
In posted
slides!!!
Principles of Econometrics, 3rd Edition
Slide B-23
Principles of Econometrics, 3rd Edition
Slide B-24
P(Y  y, X  x) f ( x, y )
f ( y | x)  P(Y  y | X  x) 

P  X  x
f X ( x)
Principles of Econometrics, 3rd Edition
y
f  y | X  3
0
.04/.18=.22
1
.14/.18=.78
(B.3)
What is the likelihood
Of having an income
If the person has a
4-year degree?
Slide B-25
P(Y  y, X  x) f ( x, y )
f ( y | x)  P(Y  y | X  x) 

P  X  x
f X ( x)
y
f  y | X  3
0
.04/.18=.22
1
Principles of Econometrics, 3rd Edition
.14/.18=.78
(B.3)
What is the likelihood
Of having an income
If the person has a
4-year degree?
78%
But only 69% in
general
Slide B-26

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.
P  Y  y | X  x   P Y  y 
(B.4)
f ( x, y)
f ( y | x) 
 fY ( y )
f X ( x)
(B.5)
f ( x, y )  f X ( x ) f Y ( y )
(B.6)
Principles of Econometrics, 3rd Edition
Slide B-27
Y = 1 if shaded Y = 0 if clear
X = numerical value (1, 2, 3, or 4)
Principles of Econometrics, 3rd Edition
Slide B-28
Principles of Econometrics, 3rd Edition
Slide B-29
Principles of Econometrics, 3rd Edition
Slide B-30
Principles of Econometrics, 3rd Edition
Slide B-31