## 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 209

then the

is MY(t) = fl MXa.t). i - 1 Proof. The

. . + a„x„)-\ — £[ea\txxea2tx2 . . . gti„tx^ = E[ea,'x']E[ea2'x2] . . . E[ea"'x"] because ...

then the

**moment**-**generating function**of where au a2, . . . , ak are real constants,is MY(t) = fl MXa.t). i - 1 Proof. The

**m.g.f.**of Y is given by MY(t) = E[e'Y) = +a2*2 + .. . + a„x„)-\ — £[ea\txxea2tx2 . . . gti„tx^ = E[ea,'x']E[ea2'x2] . . . E[ea"'x"] because ...

Page 213

Let Xu X2 be two independent gamma random variables with parameters a, = 3,

fa = 3 and x2 = 5, fa = 1, respectively. (a) Find the

What is the distribution of Yl 4.82. A certain job is completed in three steps in

series.

Let Xu X2 be two independent gamma random variables with parameters a, = 3,

fa = 3 and x2 = 5, fa = 1, respectively. (a) Find the

**m.g.f.**of Y = 2JSf, + 6X2. (b)What is the distribution of Yl 4.82. A certain job is completed in three steps in

series.

Page 243

5.3 Limiting

function of a random variable Y„ by use of the definition of limiting distribution

function obviously requires that we know F„(y) for each positive integer n. But, as

indicated ...

5.3 Limiting

**Moment**-**Generating Functions**To find the limiting distributionfunction of a random variable Y„ by use of the definition of limiting distribution

function obviously requires that we know F„(y) for each positive integer n. But, as

indicated ...

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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