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AMS 311 March 30, 2000 Homework due April 6: Chapter Six: P225: 4, 6*, 8; p232: 1, 3; p241: 2, 4, 8, 10, 16*. On to continuous distributions 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 E ( X ) xf ( x)dx, provided that the integral converges absolutely. Example c , x , is called a 1 x2 Cauchy random variable. Find c so that the f(x) is a pdf. Show that E(X) does not exist. A random variable X with density function f ( x ) Don’t be bashful about checking your old calculus books and tables of integrals! From there, you will find dx 1 x 2 arctan x. Theorem 6.2 is used to prove Theorem 6.3 (The law of the unconscious statistician). Theorem 6.2. For any continuous random variable X with probability distribution function F and density function f, 0 0 E ( X ) [1 F (t )]dt F ( t )dt . Law of the unconscious statistician. Theorem 6.3. Let X be a continuous random variable with probability density function f(x); then for any function h: RR, E (h( X )) h( x) f ( x)dx. Example: 1 , a x b, and zero otherwise be the pdf of the random variable X. b a Find E (e tX ). Let f ( x ) This theorem also is the basis for proving that expectation is a linear operator for sums of functions of X. Definition of var (X) The variance of the random variable X is still var( X ) E ( X EX ) 2 . 7.1. Uniform Distribution The distribution in the example above is called a uniform random variable over (a, b). Show that f(x) is a pdf. Find the cdf. Find E(X). Find var(X).