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L10_S10R

# L10_S10R - AMS 311 Spring Semester 2010 Lecture 10 Chapter...

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AMS 311, Spring Semester 2010 Lecture 10 Chapter Five: Continuous Random Variables Probability density function (pdf): does not give probabilities; integrate pdf to get probability. Cumulative distribution function (cdf). Relation between cdf and pdf Definition of Expected Value If X is a continuous random variable with probability density function f , the expected value of X is defined by - = dx x xf X E ) ( ) ( , provided that the integral converges absolutely. Example A random variable X with density function , , 1 ) ( 2 < < - + = x x c x f is called a Cauchy random variable. Find c so that the f(x) is a pdf. Show that E(X) does not exist. Don’t be bashful about checking your old calculus books and tables of integrals! From there, you will find + = + c x x dx arctan 1 2 . Lemma 2.1 For a nonnegative random variable Y , = 0 } { ) ( dy y Y P Y E . This lemma generalizes to the result: For any continuous random variable X with probability distribution function

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L10_S10R - AMS 311 Spring Semester 2010 Lecture 10 Chapter...

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