lecture14

lecture14 - Stat 302, Introduction to Probability Jiahua...

Info iconThis preview shows pages 1–7. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 14 January-April 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 January-April 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 (pair-wise) 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 January-April 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 (pair-wise) 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 (pair-wise) independent . Jiahua Chen () Lecture 14 January-April 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 January-April 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....
View Full Document

Page1 / 32

lecture14 - Stat 302, Introduction to Probability Jiahua...

This preview shows document pages 1 - 7. Sign up to view the full document.

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