## Introduction to mathematical statistics |

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Page 12

= 0, and the

that A1 ...

**Theorem**2. The probability of the null set is zero, that is, P(0) = 0. Proof. In**Theorem**1, take A = 0 so that A* = s/ . Accordingly, we have P(0) = 1 - P(s/) = 1-1= 0, and the

**theorem**is proved.**Theorem**3. If A1 and A2 are subsets of s/ suchthat A1 ...

Page 196

This exercise and the immediately preceding one are special instances of an

important

normal ...

This exercise and the immediately preceding one are special instances of an

important

**theorem**that will be proved in the next section. 7.4 The Central Limit**Theorem**It was seen, p. 144, that, if X1, X2, . . . , Xn is a random sample from anormal ...

Page 224

We then use the factorization

either the continuous or the discrete case the p.d.f. of Y1 is of the form gi(yi. e) = R

(yi) ^PiPWyi + nq(d)] at points of positive probability density. The points of positive

...

We then use the factorization

**theorem**. It is left as an exercise to show that ineither the continuous or the discrete case the p.d.f. of Y1 is of the form gi(yi. e) = R

(yi) ^PiPWyi + nq(d)] at points of positive probability density. The points of positive

...

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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