Chapter 17 Probability Models Binomial Probability Models Poisson Probability Models.
PROBABILITY DISTRIBUTIONSdarp.lse.ac.uk/presentations/MP2Book/OUP/ProbabilityDistributions.… ·...
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Frank Cowell : Probability distributions
PROBABILITY DISTRIBUTIONSMICROECONOMICSPrinciples and AnalysisFrank Cowell
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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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