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# Chapter 7 - CHAPTER-7 Estimation Point Estimation...

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CHAPTER-7 Estimation

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Desirable Properties of a point estimator: 1. to be unbiased for . 2. to have a small variance for large sample sizes. Definition (Unbiased): An estimator is an unbiased estimator for parameter if and only if E[ ] = . Theorem : Let be a random sample of size ‘n’ from a distribution with mean µ. The sample mean, , is an unbiased estimator for µ. ˆ ˆ ˆ ˆ n X X , ...... , 1 X

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Proof: E[ ] = E[1/n( X 1 + X 2 + X 3 +……….+ X n )] By the rules of expectations, X E[ ] = 1/n(E[X 1 ] +E[X 2 ] +E[X 3 ] +…+ E[X n ]) Since X 1 , X 2 , X 3 ,…X n constitutes a random sample of size ‘n’ from a distribution with mean µ, each of these random variables have mean µ. Hence E[ ] = 1/n(µ + µ + µ + … + µ) = 1/n(nµ) = µ n terms X X
Theorem : Let be the sample mean based on a random sample of size ‘n’ from a distribution with mean µ and variance . Then Var = /n. Var[ ] = Var[(X 1 + X 2 +…+ X n ) /n] = 1/n² [Var(X 1 ) + Var(X 2 ) + … + Var(X n )] = 1/n² [ + + … + ] = n /n² = /n. X 2 2 X 2 2 2 2 2 X

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Definition (Standard error of the mean) : Let denote the sample of size ‘n’ drawn from a distribution with standard deviation . The standard deviation of is given by / n and is called the standard error of the mean. Theorem : Let S² be the sample variance based on a random sample of size ‘n’ from a distribution with mean µ and variance . S² is an unbiased estimator for X X 2 2
Q 7.1.1: Let X

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Chapter 7 - CHAPTER-7 Estimation Point Estimation...

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