## 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 185

x = 5 if a = /?. 4.39.

) ^ - " " This demonstrates the relationship between the distribution functions of ...

**Show**that the graph of the beta p.d.f. is symmetric about the vertical line throughx = 5 if a = /?. 4.39.

**Show**, for k = 1, 2, . . . , n, that [ (t - l)UĞ - t)! * - '( ' - zr ' " dz = X (x) ^ - " " This demonstrates the relationship between the distribution functions of ...

Page 192

with the distribution having p.d.f. J\x) = e~x, 0 < x < co, zero elsewhere.

Y, = X^T2' Yl = X, + X2 + X,' Yi = Xx + X2 + Xi are mutually independent. 4.50.

**Show**that Yu Y2, Y3 are mutually independent. 4.49. Let XuX2,X3 be i.i.d., eachwith the distribution having p.d.f. J\x) = e~x, 0 < x < co, zero elsewhere.

**Show**thatY, = X^T2' Yl = X, + X2 + X,' Yi = Xx + X2 + Xi are mutually independent. 4.50.

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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