## 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 103

the joint p.d.f. f(xux2) = xi + x2, 0<x, < 1, 0<x2<1, = 0 elsewhere, cannot be written

as the product of a nonnegative function of x, alone and a nonnegative function ...

**Accordingly**, Xx and are independent. If we now refer to Example 1, we see thatthe joint p.d.f. f(xux2) = xi + x2, 0<x, < 1, 0<x2<1, = 0 elsewhere, cannot be written

as the product of a nonnegative function of x, alone and a nonnegative function ...

Page 150

+ 5 exp <72\ '— be <V _ t\ + hp — be \f(x) dx. But = exp [aM + (a]t2)/2] for all real

values of t.

by ...

**Accordingly**, M(tu t2) can be written in the form a2 t\a\{\ - p2) exp \ t2n2 - t2p — ni+ 5 exp <72\ '— be <V _ t\ + hp — be \f(x) dx. But = exp [aM + (a]t2)/2] for all real

values of t.

**Accordingly**, if we set t = tx + t2p(a2/a,), we see that M(tu t2) is givenby ...

Page 378

bound in this case is l/[«(l/0)] = 9/n. But 9/n is the variance of X. Hence X is an

efficient estimator of 9. Example 4. Let S2 denote the variance of a random

sample of ...

**Accordingly**, d\nf(X;9) d9 E(X-ey a2 e 1 e2 ~J2~¥~e The Rao-Cramer lowerbound in this case is l/[«(l/0)] = 9/n. But 9/n is the variance of X. Hence X is an

efficient estimator of 9. Example 4. Let S2 denote the variance of a random

sample of ...

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### Common terms and phrases

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 hypothesis H0 independent random variables integral joint p.d.f. Let the random Let Xu X2 limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Xu 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 simple hypothesis statistic for 9 sufficient statistic testing H0 theorem unbiased estimator variance a2 Xx and X2 Yu Y2 zero elsewhere