L11 S10 - Lecture 11 AMS 311, Spring Semester, 2010 Chapter...

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Lecture 11 AMS 311, Spring Semester, 2010 Chapter 5: Continuous Random Variables 5.1. Introduction The random variable X is continuous if there exists a nonnegative function X f , defined for all real x having the property that for any set B of real numbers . ) ( } { = B X dx x f B X P The function X f is called the probability density function (pdf). 5.2. Expectation and Variance of Continuous Random Variables Lemma 2.1 . For any nonnegative random variable Y , . } [ ) ( 0 = dy y Y P Y E Proposition 2.1 . If X is a continuous random variable with probability density function f(x) ; then for any function g : R R , E g X g x f x d x ( ( ) ) ( ) ( ) . = - ∞ 5.3. The Uniform Random Variable A random variable is said to be uniformly distributed over the interval (0,1) if its pdf is given by , 1 0 , 1 ) ( < < = x x f and zero otherwise. A random variable X is uniformly distributed over the interval ) , ( β α if its pdf is , , 1 ) ( < < - = x x f and zero otherwise. Then , 2 ) ( + = X E and . 12 ) ( ) var( 2 - = X 5.4. Normal Random Variables The random variable X is normally distributed with mean μ and variance 2 σ if its probability density function is . , 2 1 ) ( ) 2 /( ) ( 2 2 < < - = - - x e x f x X π A normal random variable with mean 0 and variance 1 is called a standard normal random variable. Its cdf is given in the equation . ) ( 2 / 2 - - = Φ x z dz e x This cdf is tabulated and is a basic reference for working problems; see page 203 of your text. All probabilities are calculated through conversion to a standard normal distribution. The tables that I will give you in the next examination and the final are below.
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Normal Probability Table, AMS 311 Examinations The table below (copied from the Actuarial series of examinations) give the value of - - = Φ x w dw e x 2 / 2 2 1
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L11 S10 - Lecture 11 AMS 311, Spring Semester, 2010 Chapter...

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