9_30_10

9_30_10 - Continuous Random Variables Probability Density...

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Continuous Random Variables Probability Density Function and Cumulative Distribution Function Andrew Liu September 30, 2010 Textbook sections: 4-1 – 4-3
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Probability Density Function Textbook sections: 4-1 – 4-3
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Comparison: Probability Mass vs. Density Function Prob. Mass Func. Prob. Density Func. Random Variable Discrete Continuous Property 1 f ( x i ) 0, i = 1 , 2 , ... f ( x ) 0 Property 2 n X i =1 f ( x i ) = 1 Z -∞ f ( x ) dx = 1 Property 3 f ( x i ) = P ( X = x i ) P ( a X b ) = Z a b f ( x ) dx P ( X = x ) = 0 Textbook sections: 4-1 – 4-3
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Probability Density Function – Zero Point Mass Probability! Note: By definition, Prob ( a X a ) = Z a a f ( x ) dx = 0! Hence, P ( x 1 X x 2 ) = P ( x 1 < X x 2 ) = P ( x 1 X < x 2 ) = P ( x 1 < X < x 2 ) . Textbook sections: 4-1 – 4-3
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Probability Density Function – Examples E.g.1 Suppose that a continuous random variable X has the probability density function of f ( x ) = x 8 , for 3 < x < 5. (a) P ( X < 4) = Textbook sections: 4-1 – 4-3
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Probability Density Function – Examples E.g.1 Suppose that a continuous random variable X has the probability density function of f ( x ) = x 8 , for 3 < x < 5. (a) P ( X < 4) = Z 4 3 x 8 dx = Textbook sections: 4-1 – 4-3
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Probability Density Function – Examples E.g.1 Suppose that a continuous random variable X has the probability density function of f ( x ) = x 8 , for 3 < x < 5. (a) P ( X < 4) = Z 4 3 x 8 dx = 1 16 x 2 ˛ ˛ ˛ ˛ x =4 x =3 = 4 2 - 3 2 16 = 0 . 4375 . (b) P ( X > 3 . 5) Textbook sections: 4-1 – 4-3
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Probability Density Function – Examples E.g.1 Suppose that a continuous random variable X has the probability density function of f ( x ) = x 8 , for 3 < x < 5. (a)
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9_30_10 - Continuous Random Variables Probability Density...

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