LEC_EAS305_F11_0921

LEC_EAS305_F11_0921 - Cumulative Distribution Functions...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychevs 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 Chebychevs Inequality Agenda for Today 1 Cumulative Distribution Functions Continuous cdfs Discrete cdfs Properties 2 Mean (Expected Value) and Variance 3 Chebychevs Inequality Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 2 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychevs Inequality Continuous cdfs Discrete cdfs 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 Chebychevs Inequality Continuous cdfs Discrete cdfs Properties Continuous cdfs 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 Chebychevs Inequality Continuous cdfs Discrete cdfs 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 Chebychevs Inequality Continuous cdfs Discrete cdfs Properties Example Example: X Exp ( ). f ( x ) = e- x if x > otherwise F ( x ) = Z x- f ( t ) dt = if x 1- e- x if x > Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 6 of 26 Cumulative Distribution Functions Mean (Expected Value) and Variance Chebychevs Inequality Continuous cdfs Discrete cdfs Properties Example Example: Flip a coin twice.Example: Flip a coin twice....
View Full Document

Page1 / 26

LEC_EAS305_F11_0921 - Cumulative Distribution Functions...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online