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Unformatted text preview: lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Outline 1 Review 2 Normal Probability Distribution 3 Computing Areas for Std Normal 4 Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Review Continuous Probability Distributions, review A continuous random variable may assume any numerical value in one or more intervals • Eg: portfolio value by the end of the year, finish time of the winning horse in a horse racing, bus waiting time,... • The probability of taking any particular value is 0, hence we use a continuous probability distribution to assign probabilities to intervals • A curve f ( x ) is the continuous probability distribution of the continuous random variable X if the probability that X will be in a specified interval is equal to the area under the curve f ( x ) corresponding to the interval • Other names for a continuous probability distribution: • Probability curve, Probability density function (pdf in short), or simply density function lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Review Area and Probability, review • For any a < b , the blue area under the curve f ( x ) from x = a to x = b is the probability that X takes value in the range a to b — denoted by P ( a ≤ X ≤ b ) • P ( a ≤ X ≤ b ) = P ( a < X < b ) = P ( a ≤ X < b ) = P ( a < X ≤ b ) , because each of the interval endpoints has probability 0 lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Review The Uniform Distribution, review X is a uniform rv on an interval [ c , d ] if X is equally like to fall in any equallength interval inside [ c , d ] • The probability curve describing the uniform distribution on an interval [ c , d ] is f ( x ) = 1 / ( d c ) if c ≤ x ≤ d otherwise • Hence the probability that X takes value between a and b ( c ≤ a < b ≤ d ) is P ( a ≤ X ≤ b ) = Area under f ( x ) from a to b = ( b a ) · 1 / ( d c ) = b a d c . lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II Review The Uniform Probability Curve, review lecture 11, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11, Continuous Random Variables II...
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This note was uploaded on 12/30/2009 for the course ISOM ISOM111 taught by Professor Anthonychan during the Fall '09 term at HKUST.
 Fall '09
 AnthonyChan

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