## 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

Moreover, CxC2= 1 because /*o0 /*0O 1 = g(x\)h{x2) dxi dx2 = gix^dxi h(x2) dx2

— C2C\- These results imply that f(xu x2) = g(xx)h(x2) = Cig(x\)c2h(x2) = f{xx)f2(

x2).

see ...

Moreover, CxC2= 1 because /*o0 /*0O 1 = g(x\)h{x2) dxi dx2 = gix^dxi h(x2) dx2

— C2C\- These results imply that f(xu x2) = g(xx)h(x2) = Cig(x\)c2h(x2) = f{xx)f2(

x2).

**Accordingly**, Xx and X2 are independent. If we now refer to Example 1, wesee ...

Page 286

(Y < 9.5) = 1 - 0.95 = 0.05, from Table II of Appendix B. When the hypothesis //, is

true, the random variable X/2 is x\2); so the random variable (Z, + X2)/2 = Z, say, ...

**Accordingly**, the power of the test when H0 is true is given by Pr (Y > 9.5) = 1 - Pr(Y < 9.5) = 1 - 0.95 = 0.05, from Table II of Appendix B. When the hypothesis //, is

true, the random variable X/2 is x\2); so the random variable (Z, + X2)/2 = Z, say, ...

Page 298

There is a way out of our trouble, however. We have noted that Qk _ i is a function

of \i and a2.

Obviously, these values depend upon the observed Xx = xu . . . , Xk = xk and are

...

There is a way out of our trouble, however. We have noted that Qk _ i is a function

of \i and a2.

**Accordingly**, choose the values of n and a2 that minimize Qk _ , .Obviously, these values depend upon the observed Xx = xu . . . , Xk = xk and are

...

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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