## Introduction to mathematical statistics |

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Page 258

The

hypothesis H1 is that function, defined for all distributions under consideration,

which yields the probability that the sample point falls in the critical region C of

the ...

The

**power function**of a test of a statistical hypothesis H0 against an alternativehypothesis H1 is that function, defined for all distributions under consideration,

which yields the probability that the sample point falls in the critical region C of

the ...

Page 260

Find the

= 2 from the distribution having p.d.f. f(x; 8) = (l/8)e~xie, 0 < x < oo, zero

elsewhere. We reject H0: 8 = 2 and accept H^. 8 = 1 if the observed values of X1,

X2, say, ...

Find the

**power function**of the test. 10.3. Let X1, X2 be a random sample of size n= 2 from the distribution having p.d.f. f(x; 8) = (l/8)e~xie, 0 < x < oo, zero

elsewhere. We reject H0: 8 = 2 and accept H^. 8 = 1 if the observed values of X1,

X2, say, ...

Page 272

Find the

the form/(x; 0) = 1/0, 0 < x < 8, zero elsewhere. Let Y1 < Y2 < Y3 < Y4 denote the

order statistics of a random sample of size 4 from this distribution. Let the

observed ...

Find the

**power function**K(8), i 0 < 8 < i, of this test. 10.17. Let X have a p.d.f. ofthe form/(x; 0) = 1/0, 0 < x < 8, zero elsewhere. Let Y1 < Y2 < Y3 < Y4 denote the

order statistics of a random sample of size 4 from this distribution. Let the

observed ...

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### Common terms and phrases

Accordingly best critical region binomial distribution cent confidence interval Chapter chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges stochastically critical region decision function defined degrees of freedom denote a random discrete type distribution having p.d.f. Equation Example EXERCISES F distribution function of Y1 given H0 is true independent random variables inequality integral joint p.d.f. Let the random Let X1 limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral order statistics Poisson distribution positive integer power function Pr X1 probability density functions probability set function quadratic form random experiment random interval random sample random variables X1 reject H0 respectively sample space Show significance level simple hypothesis H0 statistic for 9 statistic Y1 stochastically independent random subset testing H0 theorem type of random unbiased statistic variance a2 X1 and X2 Xn denote zero elsewhere