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Unformatted text preview: Estimators, Estimates, and Sampling distributions 1 Review Definition: experiment action or process that generates outcomes. Only one outcome can occur and we are usually uncertain which outcome this will be. Definition: random variable a numerical measurement of the outcome of an experiment that has yet to be performed. Definition: realization of a random variable the value that the random variable takes on after the experiment is performed. (e.g. if the random variable X counts the number of heads in 3 coin flips (we have yet to flip), and we carry-out this experiment and see 2 heads, then x = 2 is the realization of X . Definition: parameter a number that describes a probability distribution. (e.g. the mean/expected value describes the center, the standard deviation describes the spread, the probability of success for a Bernoulli or Binomial random variable affects both the center and spread) Notation We use upper-case letters for random variables (e.g. X,Y,Z ) We use lower-case letters to denote values these random variable take (e.g. x,y,z ) We use Greek letters to denote distribution parameters (e.g. X ,, X ,, ) Special types of distributions: we have seen up to now Bernoulli: Bern( ) Binomial: Binom( n, ) Normal: N ( , ) The terms distribution and probability distribution have the same meaning in this course. 1 2 The random sample 2.1 Definitions Definition: independent random variables A sequence of random variables X 1 ,...,X n are independent if P [( X 1 a 1 ) ( X 2 a 2 ) ( X n a n )] = P ( X 1 a 1 ) P ( X 2 a 2 ) P ( X n a n ) for all constants a 1 ,...,a n . Definition: independent and identically distributed A sequence of random variables X 1 ,...,X n are independent and identically distributed (iid) if X 1 ,...,X n are independent and all the X i s have the same distribution. (e.g. X i N ( = 68 , = 3) for i = 1 ,...,n ) Definition: random sample from distribution Q A sequence of random variables X 1 ,...,X n are a random sample from distribution Q if X 1 ,...,X n are independent and identically distributed, all having distribution Q . (e.g. take Q to be Bern( = 0 . 6)) Definition: realization of a random sample Let X 1 ,...,X n be a random sample from distribution Q . If we perform the experiment that generates a realization of X 1 ,...,X n , then the resulting sequence of values: x 1 ,...,x n are called a realization of a random sample from distribution Q . That is, after we performed the experiment, X 1 took on the value x 1 , X 2 took on the value x 2 , ..., X n took on the value x n . Remarks: Q represents some particular probability distribution Examples: Q could be N ( = 68 , = 4) Q could be Bern( = 0 . 9) The statistical models we will introduce in this chapter will assume that measurements we collect: x 1 ,...,x n are a realization of a random sample from some distribution, with unknown parameters:...
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- Spring '08