PostLecture11EN.pdf - Lecture 11 Chapters 8.1-8.3 Plan for...

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Lecture 11 Chapters 8.1-8.3 Plan for today I Sample distributions I Linear combination of normal distributed random variables I The distribution of the sample mean (independent, normally distributed random variables) I The distribution of the sample variance (independent, normally distributed random variables)
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Distributions of statistics I We perform n experiments, and denote the observed values by by x 1 , · · · , x n . I In our probability model the observed results are realizations of the random variables X 1 , · · · , X n . I The random variables X 1 ,..., X n form a random sample if I The random variables are X i independent , I The variables X i , i = 1 , 2 , ... are identically distributed ( have the same distribution. ) Distribution of statistics Definition A statistic is a function of X 1 , ..., X n and depends only on known constants, and therefore is a random variable. We are interested in the distribution of a statistic , to be able to estimate unknown parameters. Remark: A statistic does not depend on unknown parameters. This means that for any given set of realizations x 1 , ..., x n , one can calculate the value of the statistic.
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Examples of statistics Let X 1 , ..., X n be a random sample. The following functions of X 1 , ..., X n are examples of statistics: I X i I min { X 1 , ..., X n } I The sample mean X ( n ) = X 1 + ... + X n n I The sample variance S 2 = 1 n - 1 n X i = 1 ( X i - ¯ X ) 2 . Distribution of the sample mean I Central limit theorem: The distribution of the sample mean if n 7→ ∞ . I What can we say about the distribution of the sample mean if n is not very large?
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Linear combination of independent distributed variables. I Given X 1 , · · · , X n and constants a 1 , · · · , a n , the random variable X = n X i = 1 a i X i is a linear combination of X 1 , · · · , X n . Example: The sample mean ¯ X = n i = 1 X i n is a linear combination of X i , i = 1 , ..., n ( a i = 1 n .) Linear combination of independent normally distributed variables. If the random variables X i , i = 1 , ..., n are independent and normally distributed with expected value μ i and variance σ 2 i , then a 1 X 1 + .... + a n X n is normally distributed , with expected value n i = 1 a i μ i and variance n i = 1 a 2 i σ 2 i . J (1)
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Proof I Let X 1 , X 2 , . . . , X n be independent, normally distributed randomvariables, with expected value μ i and variance σ 2 i . Then M X ( t ) = n Y i = 1 M a i X i ( t ) . M a i X i ( t ) = E ( e a i X i t ) = M X i ( a i t ) = exp μ i a i t + a 2 i σ 2 i t 2 2 ! , and therefore M X ( t ) = n Y i = 1 M a i X i ( t ) = exp t n X i = 1 a i μ i + t 2 2 n X i = 1 a 2 i σ 2 i ! . I M X ( t ) equals the MGF of a normally distributed variable with expected value n i = 1 a i μ i and variance n i = 1 a 2 i σ 2 i I X therefore is normally distributed with expected value n i = 1 a i μ i and variance n i = 1 a 2 i σ 2 i Example 1 I A shopkeeper has 100 products in stock. The demand per day for his products is normally distributed with expected value μ = 30 and standard deviation σ = 4. Assume that the demand in different days may be modeled by independent random variables. Every three days, the shopkeeper places an order such that his inventory is replenished up to 100 products. The delivery time is 3
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