## 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 165

From this joint p.d.f. g(

on y2 or the marginal p.d.f. of Y2 by summing on y,. Perhaps it should be

emphasized that the technique of change of variables involves the introduction of

as ...

From this joint p.d.f. g(

**yu y2**) we may obtain the marginal p.d.f. of F, by summingon y2 or the marginal p.d.f. of Y2 by summing on y,. Perhaps it should be

emphasized that the technique of change of variables involves the introduction of

as ...

Page 173

The events (XuX2)e A and (

XuX2)eA] w * = h(xu x2) dx\ dx2. We wish now to change variables of integration

by writing v, = x2),y2 = u2(xu x2),orx, = w\{

The events (XuX2)e A and (

**Yu Y2**)e B are equivalent. Hence Pr[(y„ Y2)eB] = Pr[(XuX2)eA] w * = h(xu x2) dx\ dx2. We wish now to change variables of integration

by writing v, = x2),y2 = u2(xu x2),orx, = w\{

**yu y2**),x2 = w2(**yu y2**). Ithasbeen ...Page 191

Let Yx denote the mean and let Y2 denote twice the variance of the random

sample. The associated transformation is xx +x2 y\ = 2 (xi - x2f y2 — . This

transformation maps si = {(xux2): — oo<;c,<oo, — oo<x2<oo} onto St = {(

— oo < yx ...

Let Yx denote the mean and let Y2 denote twice the variance of the random

sample. The associated transformation is xx +x2 y\ = 2 (xi - x2f y2 — . This

transformation maps si = {(xux2): — oo<;c,<oo, — oo<x2<oo} onto St = {(

**yu y2**) :— oo < yx ...

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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