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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen JanuaryApril 2011 Jiahua Chen () Lecture 14 JanuaryApril 2011 1 / 26 Variance formula, again Let X and Y be two random variables. var ( aX + bY ) = a 2 var ( X ) + 2 ab cov ( X , Y ) + b 2 var ( Y ) . In particular, var ( X + Y ) = var ( X ) + 2 cov ( X , Y ) + var ( Y ) . and var ( X Y ) = var ( X ) 2 cov ( X , Y ) + var ( Y ) . When X and Y are independent, cov ( X , Y ) = 0 so we get var ( X + Y ) = var ( X ) + var ( Y ) and var ( X Y ) = var ( X ) + var ( Y ) . Note the outcome is the same whether you add to subtract. Jiahua Chen () Lecture 14 JanuaryApril 2011 2 / 26 Variance formula, again Let X and Y be two random variables. var ( aX + bY ) = a 2 var ( X ) + 2 ab cov ( X , Y ) + b 2 var ( Y ) . Additionally, var ( n i = 1 X i ) = i var ( X i ) + n i = 1 n j negationslash = i cov ( X i , X j ) . When X i s are (pairwise) independent, cov ( X i , X j ) = 0 for i negationslash = j , var ( n i = 1 X i ) = i var ( X i ) . Jiahua Chen () Lecture 14 JanuaryApril 2011 3 / 26 Variance formula, again Let X and Y be two random variables. var ( aX + bY ) = a 2 var ( X ) + 2 ab cov ( X , Y ) + b 2 var ( Y ) . Additionally, var ( n i = 1 X i ) = i var ( X i ) + n i = 1 n j negationslash = i cov ( X i , X j ) . When X i s are (pairwise) independent, cov ( X i , X j ) = 0 for i negationslash = j , var ( n i = 1 X i ) = i var ( X i ) . The validity holds only when X i s are (pairwise) independent . Jiahua Chen () Lecture 14 JanuaryApril 2011 3 / 26 Binomial distribution example What percentage of voters in BC will vote for Liberal in the next election? We may sample n voters in BC, and record the response of the i th sample by X i . It equals one or zero, if it is for or against voting liberal. The outcome of the poll helps to give us an estimate on the proportion p as p = n 1 ( x 1 + + x n ) where x i is the observed value of X i , i = 1, 2, . . . , n . Conceptually, we regard p = n 1 ( X 1 + + X n ) as an estimator of p . Jiahua Chen () Lecture 14 JanuaryApril 2011 4 / 26 Binomial distribution example It is sensible to regard each the poll is a sequence of n Bernoulli trial with probability of success p (unknown to us, but has some value between 0 and 1, not random). As an estimator p = n 1 ( X 1 + + X n ) is a random variable, and E ( p ) = p . That is, if we take the poll again and again, the estimator has average value exactly equal the target value of the parameter. Such estimators are said to be unbiased....
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 Spring '11
 Dr.Chen
 Probability, Variance

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