Lect11_L6_L7_handout1

Lect11_L6_L7_handout1 - lecture 11 w/o answer, Continuous...

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Unformatted text preview: lecture 11 w/o answer, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11 w/o answer, 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 w/o answer, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11 w/o answer, 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 w/o answer, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11 w/o answer, 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 w/o answer, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11 w/o answer, 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 equal-length 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 w/o answer, Continuous Random Variables II Review Normal Probability Distribution Computing Areas for Std Normal Connections between Std Normal and General Normals lecture 11 w/o answer, Continuous Random Variables II Review The Uniform Probability Curve, review lecture 11 w/o answer, Continuous Random Variables II Review...
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Lect11_L6_L7_handout1 - lecture 11 w/o answer, Continuous...

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