## 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 65

If the variance of the random variable X exists,

a random variable X of the continuous type have a p.d.f. f(x) whose graph is

symmetric with respect to x = c. If the mean value of X exists,

If the variance of the random variable X exists,

**show**that E(X2)>[E(X)]2. 1.93. Leta random variable X of the continuous type have a p.d.f. f(x) whose graph is

symmetric with respect to x = c. If the mean value of X exists,

**show**that E(X) = c.Page 192

with the distribution having p.d.f. f(x) = e~x, 0 < x < oo, zero elsewhere.

Y,=X^T2' Y2 = Xl + X2 + Xi' Yj-Xx + X* + Xj are mutually independent. 4.50.

**Show**that Yu Y2, F3 are mutually independent. 4.49. Let XuX2,X3 be i.i.d., eachwith the distribution having p.d.f. f(x) = e~x, 0 < x < oo, zero elsewhere.

**Show**thatY,=X^T2' Y2 = Xl + X2 + Xi' Yj-Xx + X* + Xj are mutually independent. 4.50.

Page 313

each of these unbiased estimators. 7.4. Let y, and Y2 be two independent

unbiased estimators of 9. Say the variance of Y, is twice the variance of Y2. Find

the ...

**Show**that 4 y, 2Y2, and |K3 are all unbiased estimators of 9. Find the variance ofeach of these unbiased estimators. 7.4. Let y, and Y2 be two independent

unbiased estimators of 9. Say the variance of Y, is twice the variance of Y2. Find

the ...

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Accordingly approximate best critical region bivariate normal distribution 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 gamma distribution given H0 is true hypothesis H0 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 Xu 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 testing H0 theorem u(Xu X2 unbiased estimator variance a2 XuX2 Xx and X2 Yu Y2 zero elsewhere