ENGINEER 1P03 Lecture Notes - Lecture 12: Variance, Binomial Distribution, Random Variable

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ENGINEER 1P03 Full Course Notes
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ENGINEER 1P03 Full Course Notes
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5) expectation value and variance of binomial distribution. Let (cid:1845)(cid:1866) =(cid:1850)1 +(cid:1850)2 + +(cid:1850)(cid:1866) , where (cid:1850)(cid:1861) are independently identically distributed with mean (cid:2020) If (cid:1850)1, ,(cid:1850)(cid:1866) are independently identically distributed and (cid:1850)(cid:1861) ~(cid:1840) 0,1 , then (cid:1850) = 1(cid:1866) (cid:1850)1 + +(cid:1850)(cid:1866) (cid:1866) (cid:1850)(cid:1861) (cid:1850) (cid:1861)=1 (cid:1866) 1(cid:1866) 1 (cid:1850)(cid:1861) (cid:1850) 2 (cid:1845)2 = (cid:1831)(cid:1850) =(cid:2020) (cid:1861)=1 (cid:1850) ,(cid:1845)2 are independent random variables, and (cid:1866) 1(cid:2026)2 (cid:1845)2~ (cid:1866) 1. 2 =2 (cid:1866) 1 (cid:1853),(cid:2019) ~(cid:1829)(cid:2019)(cid:1853)(cid:1872)(cid:1853) 1(cid:1857) (cid:2019)(cid:1872) ,(cid:1829) = const. 2 1 =(cid:1852)2,(cid:1852)~(cid:1840) 0,1 (cid:1845)2(cid:3409)(cid:1855)(cid:2026)2 = confidence level (in percent) Confidence intervals: (cid:1850) (cid:2020) (cid:3409) = confidence interval in percent. Let (cid:1866) =1 + +(cid:1866) be the position in an integer lattice after (cid:1866) steps, each independent with uniform probability to get to nearest neighbours; 0 =. Let (cid:1873) = (cid:1866) = for some (cid:1866) > 0 , (cid:1840) =number of visits to origin.

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