L3 Random Variables - Random Variables Generally the object...

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Random Variables Generally the object of an investigators interest is not necessarily the action in the sample space but rather some function of it. Technically a real valued function or mapping whose domain is the sample space is called a “Random Variable,” it is these that are usually the object of an investigators attention. If the mapping is onto a finite (or countably infinite) set of points on the real line, the random variable is said to be discrete. Otherwise, if the mapping is onto an uncountably infinite set of points, the random variable is continuous. This distinction is a nuisance because the nature of thing which describes the probabilistic behaviour of the random variable, called a Probability Density Function (denoted here as f(x) and referred to as a p.d.f.), will differ according to whether the variable is discrete or continuous. In the case of a discrete random variable X, with typical outcome x i , (it shall be assumed for convenience that the x i ’s are ordered with I from smallest to largest), the probability density function f(x i ) is simply the sum of the probabilities of outcomes in the sample space which result in the random variable taking on the value x i . Basically the p.d.f. for a Discrete Random Variable obeys 2 rules: i. 0 1 ) ( i x f ii. = i x possible all i x f 1 ) ( In the discrete case f(x i ) = P(X=x i ), in the continuous case it is not possible to interpret the p.d.f. in the same way indeed, since X can take on any one of an uncountably infinite set of values, we cannot give it a subscript “i”. when the random variable X is continuous with typical value x, the probability density function f(x) is a function that, when integrated over the range (a, b), will yield the probability that a sample is realized such that the resultant x would fall in that range (much as the probability function for continuous sample spaces was defined above). In this case the p.d.f. will obey three basic rules: 1. . 0 ) ( x f 2. = b a b x a P dx x f ) ( ) ( 3. = 1 ) ( dx x f Associated with these densities are Cumulative Distribution Functions F(x) which in each case yield the probability that the random variable X is less than some value x. Algebraically these may be expressed as:
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= + = = k t k k i x x x where x f x F x X P 1 1 ); ( ) ( ) ( = = x dz z f x F x X P ) ( ) ( ) ( Note that in the case of discrete random variables F(x) is defined over the whole range of the random variable and in the case of continuous distributions d(Fx)/dx = f(x) i.e. the derivative of the c.d.f. of x gives us the p.d.f. of x. Expected Values and Variances The Expected Value of a function g(x) of a random variable is a measure of its location and is defined for discrete and continuous random variables respectively as follows: = i x possible all i i x f x g X g E ) ( ) ( )) ( ( +∞ = dx x f x g X g E ) ( ) ( )) ( ( The expectations operator E( ) is simply another mathematical operator. Just as when the operator d/dx in front of a function g(x) tells us to “take the derivative of the function
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This note was uploaded on 03/23/2011 for the course ECONOMICS 209 taught by Professor Kambourov during the Spring '11 term at University of Toronto.

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L3 Random Variables - Random Variables Generally the object...

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