## Introduction to Mathematical StatisticsAn exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform. |

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Page 12

1.3. The Probability Set Function Let # denote the set of every possible outcome

of a random experiment; that is, # is the sample space. It is our purpose to define

a set function P{C) such that if C is a

1.3. The Probability Set Function Let # denote the set of every possible outcome

of a random experiment; that is, # is the sample space. It is our purpose to define

a set function P{C) such that if C is a

**subset**of #, then P{C) is the probability that ...Page 29

In such an instance we could write X(c) = c so that Let A" be a random variable

that is defined on a sample space <€, and let si be the space of X. Further, let A

be a

...

In such an instance we could write X(c) = c so that Let A" be a random variable

that is defined on a sample space <€, and let si be the space of X. Further, let A

be a

**subset**of jaf. Just as we used the terminology "the event C," with C <= #, we...

Page 30

But the reader should fully recognize that the probability set function P is defined

for

are not the same set function. Nevertheless, they are closely related and some ...

But the reader should fully recognize that the probability set function P is defined

for

**subsets**C of #, whereas Px is defined for**subsets**A of s&, and, in general, theyare not the same set function. Nevertheless, they are closely related and some ...

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Accordingly approximate best critical region chi-square distribution complete sufficient statistic conditional p.d.f. conditional probability confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random depend upon 9 discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters equation estimator of 9 Example Exercise F-distribution gamma distribution given H0 is true independent random variables integral joint p.d.f. Let the random Let Xu X2 likelihood function limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Yx percent confidence interval Poisson distribution positive integer probability density functions probability set function quadratic form random experiment random sample random variables Xx reject H0 respectively sample space Section Show significance level statistic for 9 subset testing H0 theorem u(Xu X2 unbiased estimator XuX2 Xx and X2 Yu Y2 zero elsewhere