l17_contpdf

l17_contpdf - Continuous Random Variables Reading Ross Ch 5...

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04/01/2009 CS206 - Intro. to Discrete Structures II 1 Continuous Random Variables Reading: Ross, Ch 5., Sec. 1-2. Monday, April 13, 2009
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04/01/2009 CS206 - Intro. to Discrete Structures II 2 Continuous Random Variable • X is a continuous random variable if it takes on values from the set of real numbers: X 2 < . • Examples: • Position, velocity, … of particles (objects) • Arrival time between network messages • Height of male population Monday, April 13, 2009
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04/01/2009 CS206 - Intro. to Discrete Structures II 3 Probability Density Function (pdf) f(x) (of p X (x)) is a probability density function if It is nonnegative: f(x) 0, 8 x 2 < , It is normalized to 1: s - 1 1 f(x) dx = 1. Probability of a rv X being in interval X 2 [a,b] is P(a X b ) = s a b f(x) dx. Monday, April 13, 2009
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04/01/2009 CS206 - Intro. to Discrete Structures II 4 Cumulative Distribution Function • Cumulative distribution function of a continuous rv X is F(x) = Φ (x) = P(X x) = s - 1 x f(x) dx. • Hence, pdf f(x) and cdf F(x) are related as: f(x) = dF(x)/dx. • P(a < X b ) = F(b) – F(a). Monday, April 13, 2009
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04/01/2009 CS206 - Intro. to Discrete Structures II 5 Understanding PDF Consider a small interval of length ε around value x of a cont. rv X, I = (x- ε /2,x+ ε /2). What is the probability P(X 2 I)? P(X 2 I) = s x- ε /2 x+ ε /2 f(t)dt ¼ f(x) ε . = F(x+ ε /2) – F(x- ε /2) x t f(t) x+ ε /2 x- ε /2 f(x) f(x+ ε /2) ¼ f(x) f(x- ε /2) ¼ f(x) Monday, April 13, 2009
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04/01/2009 CS206 - Intro. to Discrete Structures II 6 Example Consider random variable X 2 < , with pdf f(x) = C ¢ e -ax , x
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This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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l17_contpdf - Continuous Random Variables Reading Ross Ch 5...

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