## 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 54

If the mathematical expectation of Y exists, recall that the

dy or £ />>/g(>0 exists. Hence the existence of E[u(X)\ implies that the

corresponding

some fairly ...

If the mathematical expectation of Y exists, recall that the

**integral**(or sum) \y\g(y)dy or £ />>/g(>0 exists. Hence the existence of E[u(X)\ implies that the

corresponding

**integral**(or sum) converges absolutely. Next, we shall point outsome fairly ...

Page 131

3.3 The Gamma and Chi-Square Distributions In this section we introduce the

gamma and chi-square distributions. It is proved in books on advanced calculus

that the

a ...

3.3 The Gamma and Chi-Square Distributions In this section we introduce the

gamma and chi-square distributions. It is proved in books on advanced calculus

that the

**integral**,y»- '/,-> e y dy exists for a > 0 and that the value of the**integral**isa ...

Page 224

We shall evaluate the

then we shall subsequently set t\ = t2 = . . . = t„ = 0, and thus establish Equation (1

). First, we change the variables of integration in

...

We shall evaluate the

**integral**(x - n)'A(x - "0° expU_(l^(LzjO" dx, . . . dx„ (2) andthen we shall subsequently set t\ = t2 = . . . = t„ = 0, and thus establish Equation (1

). First, we change the variables of integration in

**integral**(2) from x, , x2, . . . , x„ to...

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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