20a_basic_prob

20a_basic_prob - Economics 251a Fall 2006 Basic Probability...

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Economics 251a Fall 2006 Basic Probability 1 States of the World De f nition Let S be a set of “states of the world.” Let γ s > 0 be the probability of each s S , so X s S γ s =1 . The pair ( S, γ ) is called a probability space. Given any subset E S ,wede f ne γ ( E ) X S E γ s . ¥ For example, let S be the set of all 36 ordered pairs ( i, j ) where i and j are integers between 1and6 .Let γ s / 36 for each s S . We might think of S as the possible displays from a roll of two dice. A picture of the dice state space is given below: 123456 1 2 3 4 5 6 Each state s corresponds to an entry in the above matrix. Let E be the states in which the f rst die shows 3. Then E = { ( i, j ): i =3 } = { (3 , 1) , (3 , 2) , (3 , 3) , (3 , 4) , (3 , 5) , (3 , 6) } .T h u s γ ( E )=6 / 36 = 1 / 6 . A random variable X on S assigns a number X s to each state s . We shall often think of X s as the payo f from asset X in state s . For example, X s could be the face of the f rst die, X ( i, j ) i for all ( i, j ) S . Similarly, Y s could be the face of the second die, Y ( i, j )= j . Notice that a state s determines the outcome of both random variables. The random variable X is constant along each row, while the random variable Y is constant down each column. We can de f ne a third random variable Z on S by Z s X s + Y s for all s S .Thu s Z is the total rolled by both dice. Notice that Z is constant along all southwest—northwest diagonals of the state space. If u : R R is any function on the reals, then u ( X ) is a random variable if x is, de f ned by u ( X ) s = u ( X s ) . Again we emphasize that each state s determines the outcomes for all random variables (such as X , Y , Z, u ( X ) )de f ned on S . 1
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2 Picturing Random Variables We can explicitly describe how Z depends on the state s by writing the value of Z s in the cell corresponding to each state s : 123 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 If the state space is large, the enumeration of all values of Z is hard to see. One simpli f cation is to concentrate on the outcome of Z , z R ( Z ) .The histogram of Z is the graph associating to each outcome z R ( Z ) its probability γ ( { s S : Z s = z } ) . For the sum of two dice, the histogram is: 123456789 1 0 1 1 1 2 1 3 8 + 6/36 5/36 4/36 3/36 2/36 1/36 0 Notice that in this histogram the highest probability is on the central value z =7 ,andthe probability falls quickly to zero as z diverges from the central value. The most famous histogram is the so-called normal bell curve f ( z )= 1 2 π e z 2 which falls doubly exponentially fast to zero as | z | →∞ . The normal histogram is said to have a “thin tail.” By contrast, the Cauchy distribution f ( z ___ has a “fat tail.” 2
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8 + 0 To give an idea of a fat tail, think of a modi f ed dice in which there were two more outcomes, 10 and 24, each with probability 1/200. Suppose that probability of Z =2 , 3 ,..., 12 was the old probability multiplied by 99/100. The histogram then would look roughly like 7 6/36 5/36 4/36 3/36 2/36 1/36 14 21 24 -10 with two points way out in the tails.
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This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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20a_basic_prob - Economics 251a Fall 2006 Basic Probability...

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