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Unformatted text preview: Chapter 4. Continuous Random Variable and Probability Distributions 4.1 Continuous Random Variable and Probability Density Functions Def) rv X is continuous if its set of possible values is an entire interval of numbers. Ex 4.1 – measure the depth of a lake at random locations minimum depth ≤ X ≤ maximum depth Ex 4.2) pH X is determined for a chemical compound 0 ≤ X ≤ 14 or 5.5 ≤ X ≤ 6.5 Def) probability distribution (probability density function; pdf) of a continuous rv is a function f(x) 220d ( ) ( ) b a P a X b f x dx ≤ ≤ = ∫ Note: height does not represent the probability. 1 Legitimate conditions for pdf: 1. ( ) 0, f x x 2200 ≥ 2. ( ) 1 f x dx ∞∞ = ∫ Ex 4.3) X angle measured clockwise to the location of an imperfection from the reference line. 1 360 ( ) 360 x f x ow ≤ < = Def) X ~ uniform distribution on the interval [A,B] if 1 ( ; , ) A x B f x A B B A ow ≤ ≤ = Prop) For continuous rv X, P(X=c)=0 2 ≤ ≤ = < ≤ = ≤ < = < < ( ) ( ) ( ) ( ) P a X b P a X b P a X b P a X b Ex 4.4) X – time headway for two cars on a freeway during a period of heavy flow. 0.15( 0.5) 0.15 0.5 ( ) x e x f x ow ≥ = .15( .5) .075 .15 0.5 .5 .15 .075 .5 ( ) .15 .15 .15 1 .15 x x x f x dx e dx e e dx e e ∞ ∞ ∞∞ ∞ = = = =  ∫ ∫ ∫ P(X ≤ 5)=? 3 4.2 Cumulative Distribution Functions and Expected Values Def) df for a continuous rv X ( ) ( ) ( ) x F x P X x f y dy∞ = ≤ = ∫ Ex 4.5 – Uniform distribution on [A,B] 1 ( ) x A f x A x B B A x B < = ≤ ≤  1 ( ) ( ) x x A x A F x f y dy dy B A B A∞ = = = ∫ ∫ 4 Prop) ( ) 1 ( ) P X a F a = ( ) ( ) ( ) P a X b F b F a ≤ ≤ = Ex 4.6) dynamic load on a bridge 1 3 2 ( ) 8 8 x x f x ow + ≤ ≤ = 2 3 ( ) 2 8 16 1 2 x x F x x x x < = + ≤ ≤ < 5 (1 1.5) ? ( 1) ? P X P X ≤ ≤ = = Obtaining f(x) from F(x) Prop) ( ) ( ) ( ) d f x F x F x dx ′ = = Ex 4.7) – Ex 4.5 1 when '( ) ( ) ( ) when x<A or x>B, '( ) ( ) d x A A x B F x f x dx B A B A F x f x ≤ ≤ = = = = = Percentile 85 th percentile – 85% of all population scores are below and...
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 Fall '08
 BUD
 Statistics, Normal Distribution, Probability, ex, η

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