An introduction to probability theory and its applications by William Feller

By William Feller

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The estimate for E{T{6)\0) is f* = j^Yli ^i ^^^ ^^ estimate for Var(T(^)|^) is j^ J2{Ti — f*)'^. 4. Finally, classical statistics has come up with many new methods for dealing with high-dimensional problems. A couple of them will be discussed in Chapter 9. 7 Exercises 1. Verify that A^(/i, cr^), exponential with f{x\6) = | e ~ ^ / ^ , Bernouni(p), binomial B(n,p), and Poisson 'P(A), each constitutes an exponential family. 2. 4). 3. 4), show that ^ > 0. 4. (a) Generate data by drawing a sample of size n = 30 from A/'(/i, 1) with /JL = 2.

Some scientists and philosophers, notably Jeffreys and Carnap, have argued that there may be a third kind of probability that applies to scientific hypotheses. It may be called objective or conventional or non-subjective in the sense that it represents a shared belief or shared convention rather than an expression of one person's subjective uncertainty. Fortunately, the probability calculus remains the same, no matter which kind of probability one uses. A Bayesian takes the view that all unknown quantities, namely the unknown parameter and the data before observation, have a probability distribution.

Cox (1958)) To estimate /i in N{ii^a'^)^ toss a fair coin. Have a sample of size n = 2 if it is a head and take n = 1000 if it is a tail. An unbiased estimate of// is Xn = X^ILi ^ ^ / ^ with variance = | { ^ + ^ } "^ x * Suppose it was a tail. Would you believe (j^/4 is a measure of accuracy of the estimate? 6. d. I7(l9-^,6>4-|). Let X±C be a 95% confidence interval, C > 0 being suitably chosen. Suppose Xi == 2 and X2 = 1. Then we know for sure 0 = (Xi-hX2)/2 and hence 0 e {X-C, X-\-C). Should we still claim we have only 95% confidence that the confidence interval covers 01 One of us (Ghosh) learned of this example from a seminar of D.

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