PROBABILITY DISTRIBUTIONS
MICROECONOMICS
Principles and Analysis
Frank Cowell
July 2015
Frank Cowell : Probability distributions
1
Purpose
 This presentation concerns statistical distributions in
microeconomics
• a brief introduction
• it does not pretend to generality
 Distributions make regular appearances in
• models involving uncertainty
• representation of aggregates
• strategic behaviour
• empirical estimation methods
 Certain concepts and functional forms appear regularly
 This presentation focuses on
• essential concepts for economics
• practical examples
July 2015
Frank Cowell : Probability distributions
2
Ingredients of a probability model
 The variate
• could be a scalar – income, family size…
• could be a vector – basket of consumption, list of inputs
 The support of the distribution
• the smallest closed set W whose complement has probability zero
• convenient way of specifying what is logically feasible (points in the
support) and infeasible (other points)
• important to check whether support is bounded above / below
 Distribution function F
• represents probability in a convenient and general way
• from this get other useful concepts
• use F for both discrete and continuous distributions
July 2015
Frank Cowell : Probability distributions
3
Types of distribution
 Discrete distributions
• W consists of a finite, or countably infinite, set of points
• F(x) takes the form of a step function
• let’s assume that support is a finite set (x1, x2,…, xn)
• distribution given as a probability vector (p1, p2,…, pn)
• E x = p1 x1 + p2 x2 +…+ pn xn
 Continuous distributions
• for univariate distributions W is usually an interval on the real
line 𝑥, 𝑥
• if F is differentiable on W then f(x), the derivative of F(x), is
known as the density at point x
• Ex =
July 2015
𝑥
𝑥d𝐹
𝑥
𝑥 =
𝑥
𝑥𝑓(𝑥)d𝑥
𝑥
a collection of
examples
Frank Cowell : Probability distributions
4
Some examples
 Begin with two cases of discrete distributions
• #W = 2. Probability p of value x0; probability 1 – p of value x1
• #W = 5. Probability pi of value xi, i = 0,…,4
 Then a simple example of continuous distribution with
bounded support
• The rectangular distribution – uniform density over an
interval
 Finally an example of continuous distribution with
unbounded support
July 2015
Frank Cowell : Probability distributions
5
Discrete distribution: Example 1
Suppose of x0 and x1 are the only
possible values
Below x0 probability is 0
Probability of x ≤ x0 is p
1
Probability of x ≥ x0 but less than x1 is p
Probability of x ≤ x1 is 1
F(x)
p
x
x0
July 2015
x1
Frank Cowell : Probability distributions
6
Discrete distribution: Example 2
There are five possible values: x0 ,…, x4
Below x0 probability is 0
Probability of x ≤ x0 is p0
1
Probability of x ≤ x1 is p0+p1
Probability of x ≤ x2 is p0+p1 +p2
Probability of x ≤ x3 is p0+p1+p2+p3
p0+p1+p2+p3
p0+p1+p2
Probability of x ≤ x4 is 1
F(x)
p0+p1
 p0 + p1+ p2+ p3+ p4 = 1
p0
x
x0
July 2015
x1
x2
x3
x4
Frank Cowell : Probability distributions
7
“Rectangular” : density function
Suppose values are uniformly
distributed between x0 and x1
Below x0 probability is 0
f(x)
x
x0
July 2015
x1
Frank Cowell : Probability distributions
8
Rectangular distribution
Values are uniformly
distributed over the
interval [x0 , x1]
1
Below x0 probability is 0
Probability of x ≥ x0 but less
than x1 is [x  x0 ] / [x1  x0]
F(x)
Probability of x ≤ x1 is 1
x
x0
July 2015
x1
Frank Cowell : Probability distributions
9
Lognormal density
Support is unbounded above
The density function with
parameters m = 1, s = 0.5
The mean
x
0
July 2015
1
2
3
4
5
6
7
8
9
10
Frank Cowell : Probability distributions
10
Lognormal distribution function
1
x
0
July 2015
1
2
3
4
5
6
7
8
9
10
Frank Cowell : Probability distributions
11