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Chapter 4 - Chapter 4 Continuous Random Variable and...

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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
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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 0 360 ( ) 360 0 x f x ow < = Def) X ~ uniform distribution on the interval [A,B] if 1 ( ; , ) 0 A x B f x A B B A ow = - Prop) For continuous rv X, P(X=c)=0 2
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= < = < = < < ( ) ( ) ( ) ( ) 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 ( ) 0 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
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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] 0 1 ( ) 0 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
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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 0 2 ( ) 8 8 0 x x f x ow + = 2 0 0 3 ( ) 0 2 8 16 1 2 x x F x x x x < = + < 5
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(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, '( ) 0 ( ) 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 15% are above that score Def) ( ) p η = (100p)th percentile - ( ) p η
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