## 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

The

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 ...

The

**Probability Set Function**Let %> denote the set of every possible outcome ofa 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 18

For illustration, this means that if C = {1,2,3}, then 3 12 3 6 2 Whether this

experiment a large number of times. EXERCISES 1.17. A positive integer from

one to ...

For illustration, this means that if C = {1,2,3}, then 3 12 3 6 2 Whether this

**probability set function**is realistic can only be checked by performing the randomexperiment a large number of times. EXERCISES 1.17. A positive integer from

one to ...

Page 21

But these are precisely the conditions that a

Accordingly, P(C2|C,) is a

may be called the conditional

; ...

But these are precisely the conditions that a

**probability set function**must satisfy.Accordingly, P(C2|C,) is a

**probability set function**, defined for subsets of C\ . Itmay be called the conditional

**probability set function**, relative to the hypothesis C,; ...

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Accordingly approximate best critical region bivariate normal distribution 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 gamma distribution given H0 is true hypothesis H0 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 Xu 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 simple hypothesis statistic for 9 testing H0 theorem u(Xu X2 unbiased estimator variance a2 XuX2 Xx and X2 Yu Y2 zero elsewhere