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# JMF-5-07 - CH Reilly UCF IEMS STA 3032 1 Continuous Random...

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Unformatted text preview: CH Reilly UCF - IEMS - STA 3032 1 Continuous Random Variables Chapter 5 Note Set 5 Spring 2007 CH Reilly UCF - IEMS - STA 3032 2 Continuous random variable A random variable is a function that assigns an (unknown) numerical value from a sample space to each possible outcome or trial. Discrete random variable – countable number of possible values. (Chapter 4) Continuous random variable – uncountable number of possible values. (Chapter 5) The probability distribution for a continuous random variable may be represented in functional or graphical form. CH Reilly UCF - IEMS - STA 3032 3 Probability density function (pdf) ∫ ∫ = < < = ≤ ≤ = 2200 ≥ ∞ ∞- b a dx x f b X a b X a dx x f x x f ) ( ) Pr( ) Pr( 1 ) ( ) ( CH Reilly UCF - IEMS - STA 3032 4 Important distinctions ) ( ) Pr( ) Pr( x f x X x x X ≠ = 2200 = = CH Reilly UCF - IEMS - STA 3032 5 Cumulative distribution function (cdf) ∫ ∫ =- = ≤ < = = ≤ = ∞- b a x dx x f a F b F b X a x f dx x dF dt t f x X x F ) ( ) ( ) ( ) Pr( ) ( ) ( ) ( ) Pr( ) ( CH Reilly UCF - IEMS - STA 3032 6 Expectation and variance ( 29 2 2 2 2 2 2 ) ( E ) ( E ) ( Var ) ( ) ( E ) ( ) ( E σ σ σ μ + =- = = = = = ∫ ∫ ∞ ∞- ∞ ∞- X X X dx x f x X dx x xf X CH Reilly UCF - IEMS - STA 3032 7 More on expectation, variance ) ( Var ) ( Var ) ( Var ) ( Var ) ( Var ) ( E ) ( E ) ( E ) ( E ) ( E 2 X c cX X X c c X c cX X c X c c c = = + = = + = + = CH Reilly UCF - IEMS - STA 3032 8 Example ≤ ≤- = otherwise 1 for ) 2 ( 3 / 2 ) ( x x x f CH Reilly UCF - IEMS - STA 3032 9 Example (cont’d.) ≤ ≤ <- =- = - =- ∫ 1 1 1 3 / ) 4 ( ) ( 3 4 2 2 3 2 ) 2 ( 3 2 : cdf 2 2 2 x x x x x x F x x t t dt t x x CH Reilly UCF - IEMS - STA 3032 10 Example (cont’d.) ) 42 . Pr( 55 . 13 . 68 . ) 1 . ( ) 6 . ( ) 6 . 1 . Pr( 48 . 3 ) 4 . ( ) 4 . ( 4 ) 4 . ( 2 = = =- =- = ≤ ≤ =- = X F F X F CH Reilly UCF - IEMS - STA 3032 11 Example (cont’d.) 2 8 3 3 . 1 6 2 / 1 3 0 8 2 5 . 1 6 2 1 3 9 4 1 8 5 ) ( V ar 1 8 5 ) 2 ( 3 2 ) ( E 9 4 ) 2 ( 3 2 ) ( E 2 2 1 2 2 1 ≈ = ≈ = - = = = =- = = =- = = ∫ ∫ σ σ μ X d x x x X d x x x X CH Reilly UCF - IEMS - STA 3032 12 Normal random variable The normal random variable is arguably the most important of all the classical (or named) random variables. There are many applications of normal random variables, particularly in inferential statistics. Normal distributions have a bell or mound shape. A unique feature of the normal random variable is that the mean, median, and mode are equal. CH Reilly UCF - IEMS - STA 3032 13 Normal pdf ∞ < < ∞- = = =-- x e x f X X x 2 1 ) , ; ( ) ( Var ) ( E 2 2 2 ) ( 2 2 σ μ π σ σ μ σ μ CH Reilly UCF - IEMS - STA 3032 14 Normal cdf We cannot integrate the normal pdf analytically....
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JMF-5-07 - CH Reilly UCF IEMS STA 3032 1 Continuous Random...

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