## 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 228

Let X,,X2, . . . ,X„ have a multivariate normal distribution with positive definite

covariance

independent if and only if V is a diagonal

pa paxa2 a\ ...

Let X,,X2, . . . ,X„ have a multivariate normal distribution with positive definite

covariance

**matrix**V. Prove that these random variables are mutuallyindependent if and only if V is a diagonal

**matrix**. 4.130. Let « = 2 and take V = a\pa paxa2 a\ ...

Page 485

Let A be a real symmetric

numbers of A is equal to 1 if and only if A2 = A. Hint: Let L be an orthogonal

only if ...

Let A be a real symmetric

**matrix**. Prove that each of the nonzero characteristicnumbers of A is equal to 1 if and only if A2 = A. Hint: Let L be an orthogonal

**matrix**such that L'AL = diag [aua2, . . . , a„] and note that A is idempotent if andonly if ...

Page 486

Let A = [a0] be a real symmetric

the squares of the characteristic numbers of A. Hint: If L is an orthogonal

show that £ £ a// = tr (A2) = tr (L'A2L) = tr [(L'AL)(L'AL)]. > ' 10.41. Let X and S2 ...

Let A = [a0] be a real symmetric

**matrix**. Prove that £ £ a2 is equal to J i the sum ofthe squares of the characteristic numbers of A. Hint: If L is an orthogonal

**matrix**,show that £ £ a// = tr (A2) = tr (L'A2L) = tr [(L'AL)(L'AL)]. > ' 10.41. Let X and S2 ...

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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