## 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

Accordingly, we say that E[u(X)] is the expectation (mathematical expectation or

expected value) of u(X). Remark. If the mathematical expectation of Y exists,

recall that the

of ...

Accordingly, we say that E[u(X)] is the expectation (mathematical expectation or

expected value) of u(X). Remark. If the mathematical expectation of Y exists,

recall that the

**integral**(or sum) \y\g(y) dy or £ MsOO exists. Hence the existenceof ...

Page 131

Why? 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

Why? 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**yx-\e~ydy exists for a > 0 and that the value of the**integral**is a ...Page 224

t„ = 0, and thus establish Equation (1). First, we change the variables of

integration in

where y' = [yu y2, . . . , yn]. The Jacobian of the transformation is one and the «-

dimensional ...

t„ = 0, and thus establish Equation (1). First, we change the variables of

integration in

**integral**(2) from x,, x2, . . . , x„ to >', , y2, . . . , y„ by writing x - fi = y,where y' = [yu y2, . . . , yn]. The Jacobian of the transformation is one and the «-

dimensional ...

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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 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