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POL571 Lecture Notes: Expectation and Functions of Random ...

POL 571: Expectation and Functions of RandomVariablesKosuke ImaiDepartment of Politics, Princeton UniversityMarch 10, 20061 Expectation and IndependenceTo gain further insights about the behavior of Random variables, we first consider theirexpectation,which is also calledmean valueorexpected value. The definition of Expectation follows our 1 LetXbe a Random variable andgbe any IfXis discrete, then the Expectation ofg(X)is defined as, thenE[g(X)] = x Xg(x)f(x),wherefis the probability mass function ofXandXis the support IfXis continuous, then the Expectation ofg(X)is defined as,E[g(X)] = g(x)f(x)dx,wherefis the probability density function (X) = orE(X) = ( ,E(|X|) = ), then we say the expectationE(X) does not sometimes writeEXto emphasize that the Expectation is taken with respect to a particularrandom variableX. For a continuous Random variable , the Expectation is sometimes written as,E[g(X)] = x g(x)d F(x).whereF(x) is the distribution function ofX.

8. Cauchy distribution. A Cauchy random variable takes a value in (−∞,∞) with the fol-lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2]. The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.

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