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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. μ X ,μ,σ X ,σ,θ ) Special types of distributions: we have seen up to now • Bernoulli: Bern( θ ) • Binomial: Binom( n,θ ) • Normal: N ( μ,σ ) 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 ( μ = 68 ,σ = 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( θ = 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 ( μ = 68 ,σ = 4) – Q could be Bern( θ = 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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This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.
- Spring '08