Frank Cowell : Probability distributions

PROBABILITY DISTRIBUTIONSMICROECONOMICSPrinciples and AnalysisFrank Cowell

April 2018 1

Frank Cowell : Probability distributions

Purpose

§ 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

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Frank Cowell : Probability distributions

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

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Frank Cowell : Probability distributions

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) • Ex = p1 x1 + p2 x2 +…+ pn xn

§Continuous distributions• for univariate distributions W is usually an interval on the real

line !, !• if F is differentiable on W then f(x), the derivative of F(x), is

known as the density at point x

• Ex =∫$$ !d& ! =∫$

$ !((!)d! a collection of examples

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Frank Cowell : Probability distributions

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

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Frank Cowell : Probability distributions

Discrete distribution: Example 1

x

§Below x0 probability is 0§Probability of x ≤ x0 is p

x1x0

1

p

§Probability of x ≤ x1 is 1

§Suppose x0, x1 are the only possible values

F(x)

§Probability of x ≥ x0 but less than x1 is p

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Frank Cowell : Probability distributions

Discrete distribution: Example 2

x

§Below x0 probability is 0§Probability of x ≤ x0 is p0

x1x0

1

p0

§Probability of x ≤ x1 is p0+p1

§There are five possible values: x0 ,…, x4

F(x)p0+p1

p0+p1+p2+p3

§Probability of x ≤ x2 is p0+p1 +p2

x4x2 x3

p0+p1+p2

§Probability of x ≤ x3 is p0+p1+p2+p3

§Probability of x ≤ x4 is 1

§ p0 + p1+ p2+ p3+ p4 = 1

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Frank Cowell : Probability distributions

“Rectangular” : density function

x

§Below x0 probability is 0

x1x0

§Suppose values are uniformly distributed between x0 and x1

f(x)

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Frank Cowell : Probability distributions

Rectangular distribution

x

§Below x0 probability is 0§Probability of x ≥ x0 but less than x1is [x - x0 ] / [x1 - x0]

x1x0

1

§Probability of x ≤ x1 is 1

§Values are uniformly distributed over the interval [x0 , x1]

F(x)

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Frank Cowell : Probability distributions

Lognormal density

x0 1 2 3 4 5 6 7 8 9 10

§Support is unbounded above§The density function with parameters µ = 1, s = 0.5

§The mean

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Frank Cowell : Probability distributions

Lognormal distribution function

x0 1 2 3 4 5 6 7 8 9 10

1

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