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 fi 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 , we de fi 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 1 and 6. Let γ s = 1 / 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: 1 2 3 4 5 6 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 fi rst die shows 3. Then E = { ( i, j ) : i = 3 } = { (3 , 1) , (3 , 2) , (3 , 3) , (3 , 4) , (3 , 5) , (3 , 6) } . Thus 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 ff from asset X in state s . For example, X s could be the face of the fi 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 fi ne a third random variable Z on S by Z s X s + Y s for all s S . Thus 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 fi 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 fi 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 : 1 2 3 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 fi 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: 1 2 3 4 5 6 7 8 9 10 11 12 13 8 + 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 , and the 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 + 8 0 To give an idea of a fat tail, think of a modi fi 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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