slides15 - Lecture Stat 302 Introduction to Probability -...

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Lecture Stat 302 Introduction to Probability - Slides 15 AD March 2010 AD () March 2010 1 / 18
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Continuous Random Variable Let X a (real-valued) continuous r.v. . It is characterized by its pdf f : R ! [ 0 , ) which such that for any set A of real numbers P ( X 2 A ) = Z A f ( x ) dx . and its distribution function F ( x ) = Pr ( X x ) = Z x ± f ( y ) dy . For any real-valued function g : R ! R , we have E ( g ( X )) = Z ± g ( x ) f ( x ) dx AD () March 2010 2 / 18
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Normal Random Variables Also known as Gaussian random variables in the literature. We say that X is a normal r.v. of parameters μ , σ 2 ± if its pdf is f ( x ) = 1 p 2 πσ exp ( x μ ) 2 2 σ 2 ! . The normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean; e.g. the heights of USA males are roughly normally distributed. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used. AD () March 2010 3 / 18
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Properties of Normal Random Variables It can indeed be checked that Z exp ( x μ ) 2 2 σ 2 ! dx = p 2 πσ . We have also E ( X ) = μ and Var ( X ) = σ 2 Hence μ is referred to as the mean and σ 2 as the variance. AD () March 2010 4 / 18
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We have P ( μ σ ± X ± μ + σ ) ² 0 . 68 , P ( μ 2 σ ± X ± μ + 2 σ ) ² 0 . 95 , P ( μ 3 σ ± X ± μ + 3 σ ) ² 0 . 997 . This helps doing quickly some approximate calculations. The distribution of the scores of the more than 1.3 million high school seniors in 2002 who took the SAT verbal exam is close to normal with μ , σ 2 ± = ( 504 , 111 2 ) . Hence 95% of the SAT scores are between 504
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slides15 - Lecture Stat 302 Introduction to Probability -...

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