Lecture 06-continuous probability distributions

Lecture 06-continuous probability distributions - 1036:...

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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 6 Lecture 6 Some Continuous Some Continuous Probability Distributions Probability Distributions
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-2 Continuous Uniform Distribution • A flat density function defined on a closed interval. • The density function of continuous uniform random variable X on the interval [ A , B ] is elsewhere , 0 , 1 ) , ; ( B x A A B B A x f = Mean: Variance: () 2 B A + = µ ( ) 12 2 2 A B = σ Constant weight Proof.
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-3 Normal (Gaussian) Distribution • The density function of the normal random variable X , with mean µ & variance σ 2 , is () . , 2 exp 2 1 ) , ; ( 2 2 < < = x x x n σ µ π A function depends on µ and σ Mean: µ Variance: σ 2 Proof.
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-4 Remarks Normal curve is dependent on the mean and standard deviation
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-5 Properties for Normal Distribution • The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x = µ . • The curve is symmetric about a vertical axis through the mean µ . • The curve has its points of inflection at x = µ±σ . (The curve is concave downward if µ−σ < X < µ + σ ; and the curve is concave upward otherwise) • The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean. • The total area under the curve and above the horizontal axis is equal to 1.
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-6 Area under the Normal Curve () ( ) = = 2 1 2 1 2 2 2 1 2 exp 2 1 , ; x x x x dx x dx x n x X x P σ µ π The two shaded regions are different in size Normal curve is dependent on the mean and standard deviation
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Prob. & Stat. Lecture06 - continuous probability distribution (cwliu@twins.ee.nctu.edu.tw) 6-7 Standard Normal Distribution • Consider the area under the normal curve, () ( ) ( ) = = 2 1 2 1 2 2 2 1 2 exp 2 1 , ; x x x x dx x dx x n x X x P σ µ π σ dz dx x z = =
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This note was uploaded on 08/23/2009 for the course IEE 1036 taught by Professor Cwliu during the Spring '06 term at National Chiao Tung University.

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Lecture 06-continuous probability distributions - 1036:...

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