1. Frank Cowell : Probability distributions PROBABILITY DISTRIBUTIONS MICROECONOMICS Principles and Analysis Frank Cowell July 2015 1

2. Frank Cowell : Probability distributions 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 2

3. 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  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 3

4. Frank Cowell : Probability distributions Types of distribution a collection of examples July 2015 4

5. Frank Cowell : Probability distributions Some examples  Begin with two cases of discrete distributions #  = 2. Probability  of value x 0 ; probability 1 –  of value x 1 #  = 5. Probability  i of value x i, 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 5

6. Frank Cowell : Probability distributions Discrete distribution: Example 1 x  Below x 0 probability is 0  Probability of x ≤ x 0 is  x1x1 x0x0 1   Probability of x ≤ x 1 is 1  Suppose of x 0 and x 1 are the only possible values F(x)  Probability of x ≥ x 0 but less than x 1 is  July 2015 6

7. Frank Cowell : Probability distributions Discrete distribution: Example 2 x  Below x 0 probability is 0  Probability of x ≤ x 0 is   x1x1 x0x0 1 00  Probability of x ≤ x 1 is   +    There are five possible values: x 0,…, x 4 F(x)  0  1  0  1  2  3  Probability of x ≤ x 2 is   +   +   x4x4 x2x2 x3x3  0  1  2  Probability of x ≤ x 3 is   +   +   +    Probability of x ≤ x 4 is 1    +   +   +   +   = 1 July 2015 7

8. Frank Cowell : Probability distributions “Rectangular” : density function x  Below x 0 probability is 0 x1x1 x0x0  Suppose values are uniformly distributed between x 0 and x 1 f(x) July 2015 8

9. Frank Cowell : Probability distributions Rectangular distribution x  Below x 0 probability is 0  Probability of x ≥ x 0 but less than x 1 is [x  x 0 ] / [x 1  x 0 ] x1x1 x0x0 1  Probability of x ≤ x 1 is   Values are uniformly distributed over the interval [x 0, x 1 ] F(x) July 2015 9

10. Frank Cowell : Probability distributions Lognormal density x 012345678910  Support is unbounded above  The density function with parameters  = 1,  = 0.5  The mean July 2015 10

11. Frank Cowell : Probability distributions Lognormal distribution function x 012345678910 1 July 2015 11