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cltnotes - Distribution of the Sample Mean and Linear...

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Distribution of the Sample Mean and Linear Combinations – De…nitions/Notes De…nition 1 A statistic is any quantity whose value can be calculated from sample data. A statistic is a random variable and will be denoted by an uppercase letter; a lowercase letter is used to represent the calculated of observed value of the statistic. De…nition 2 The random variables X 1 , X 2 , ::: , X n are said to form a simple random sample of size n if the X i ’s are independent random variables and every X i has the same probability distribution. Proposition 1 Let X 1 , X 2 , ::: , X n be a random sample from a distribution with mean value ¹ and standard deviation ¾ . Then: 1. E ( ¹ X ) = ¹ ¹ X = ¹: 2. V ( ¹ X ) = ¾ 2 ¹ X = ¾ 2 n and ¾ ¹ X = ¾ p n : In addition, with T o = X 1 + X 2 + ¢¢¢ + X n (the sample total), E ( T o ) = , V ( T o ) = 2 , and ¾ T o = ¾ p n . Proposition 2 Let X 1 , X 2 , ::: , X n be a random sample from a normal distribution with mean ¹ and standard deviation
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cltnotes - Distribution of the Sample Mean and Linear...

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