## 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

given by g(

where x, = WiO^, v2), x2 = w2(.y,,.y2) is the single-valued inverse of v, = «,(x,, x2),

...

**p.d.f.**of the two new random variables**Yx**= ux(XuX2) and Y2 = u2(Xx , X2) isgiven by g(

**yx**» yi) = Rwx (**yx**, y2), w2(**yx**,y2)], ( v, , y2) e <#, = 0 elsewhere,where x, = WiO^, v2), x2 = w2(.y,,.y2) is the single-valued inverse of v, = «,(x,, x2),

...

Page 178

Since the joint

elsewhere, the joint p.d.f. of Xx and X2 is / x] + x\\ ... We close this section by

observing a way of finding the p.d.f. of a sum of two independent random

variables. Let Xx and ...

Since the joint

**p.d.f. of Yx**and Y2 is 1 on 0 < y, < 1, 0 < y2 < 1, and zeroelsewhere, the joint p.d.f. of Xx and X2 is / x] + x\\ ... We close this section by

observing a way of finding the p.d.f. of a sum of two independent random

variables. Let Xx and ...

Page 179

Let y, = X\ + X2 and Y2 = X\- X2. Find the joint

these random variables are independent. 4.31. Let A", and X2 denote a random

sample of size 2 from a distribution that is N(n, a2). Let Yx = Xx + X2 and Y2 = Xx

+ ...

Let y, = X\ + X2 and Y2 = X\- X2. Find the joint

**p.d.f. of Yx**and y2 and show thatthese random variables are independent. 4.31. Let A", and X2 denote a random

sample of size 2 from a distribution that is N(n, a2). Let Yx = Xx + X2 and Y2 = Xx

+ ...

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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