LEC_EAS305_F11_0921

# LEC_EAS305_F11_0921 - Cumulative Distribution Functions...

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Unformatted text preview: Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Fall 2011 EAS 305 Lecture Notes Prof. Jun Zhuang University at Buffalo, State University of New York September 21, ... 2011 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 1 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Agenda for Today 1 Cumulative Distribution Functions Continuous cdf’s Discrete cdf’s Properties 2 Mean (Expected Value) and Variance 3 Chebychev’s Inequality Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 2 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Continuous cdf’s Discrete cdf’s Properties Definition Definition: For any RV X , the cumulative distribution function (cdf) is defined for all x by F ( x ) ≡ P ( X ≤ x ). X continuous implies F ( x ) = Z x-∞ f ( t ) dt . X discrete implies F ( x ) = X { y | y ≤ x } f ( y ) = X { y | y ≤ x } Pr( X = y ) . Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 3 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Continuous cdf’s Discrete cdf’s Properties Continuous cdf’s Theorem If X is a continuous RV, then f ( x ) = F ( x ) . Proof: F ( x ) = d dx R x-∞ f ( t ) dt = f ( x ), by the fundamental theorem of calculus. Remark: If X is continuous, you can get from the pdf f ( x ) to the cdf F ( x ) by integrating. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 4 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Continuous cdf’s Discrete cdf’s Properties Example Example: X ∼ U (0 , 1). f ( x ) = 1 if 0 < x < 1 0 otherwise F ( x ) = 0 if x ≤ x if 0 < x < 1 1 if x ≥ 1 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 5 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Continuous cdf’s Discrete cdf’s Properties Example Example: X ∼ Exp ( λ ). f ( x ) = λ e- λ x if x > otherwise F ( x ) = Z x-∞ f ( t ) dt = if x ≤ 1- e- λ x if x > Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 6 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychev’s Inequality Continuous cdf’s Discrete cdf’s Properties Example Example: Flip a coin twice.Example: Flip a coin twice....
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## This note was uploaded on 09/25/2011 for the course ECON 101 taught by Professor Wang during the Spring '11 term at SUNY Buffalo.

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LEC_EAS305_F11_0921 - Cumulative Distribution Functions...

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