## 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 134

For no obvious reason, we call the parameter r the number of degrees of freedom

of the

For no obvious reason, we call the parameter r the number of degrees of freedom

of the

**chi**-**square distribution**(or of the chi-square p.d.f.). Because the**chi**-**square****distribution**has an important role in statistics and occurs so frequently, we write ...Page 484

Now that we have found, in suitable form, the m.g.f. of our random variable, let us

turn to the question of the conditions that must be imposed if X'AX/a2 is to have a

Now that we have found, in suitable form, the m.g.f. of our random variable, let us

turn to the question of the conditions that must be imposed if X'AX/a2 is to have a

**chi**-**square distribution**. Assume that X'AX/a2 is x\k). Then M(t) = [(1 - 2fa,)(l ...Page 562

452, 467 Limiting distribution, 233, 237, 243, 253, 294, 380 Limiting moment-

generating function, 243 Linear ... 486, 510 of binomial distribution, 1 18 of

bivariate normal distribution, 149 of

distribution, ...

452, 467 Limiting distribution, 233, 237, 243, 253, 294, 380 Limiting moment-

generating function, 243 Linear ... 486, 510 of binomial distribution, 1 18 of

bivariate normal distribution, 149 of

**chi**-**square distribution**, 134 of gammadistribution, ...

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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 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