## Introduction to mathematical statistics |

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Page 69

First the product of two nonnegative functions /1(

is positive on a product space. That is, if f1(

only on, the respective spaces s/1 and si2, then the product of f1(xj) and f2(

...

First the product of two nonnegative functions /1(

**x1**)/2(**x2**) means a function thatis positive on a product space. That is, if f1(

**x1**) and f2(**x2**) are positive on, andonly on, the respective spaces s/1 and si2, then the product of f1(xj) and f2(

**x2**) is...

Page 70

Proof. If

where f^Xj) and /2(x2) are the marginal probability density functions of

respectively. Thus, the condition f(

...

Proof. If

**X1 and X2**are stochastically independent, then f(**x1**,**x2**) = f1(x1)f2(x2),where f^Xj) and /2(x2) are the marginal probability density functions of

**X1 and X2**respectively. Thus, the condition f(

**x1**,**x2**) s g(xi)&(x2) is fulfilled. Conversely, if /(*j...

Page 76

A probability model for each X, is the p.d.f. f(x) = x = 0, 1, zero elsewhere. Since

the trials are independent, we say that

independent. Then, for example, the probability that the first head appears on the

...

A probability model for each X, is the p.d.f. f(x) = x = 0, 1, zero elsewhere. Since

the trials are independent, we say that

**X1**,**X2**, X3, . . . are mutually stochasticallyindependent. Then, for example, the probability that the first head appears on the

...

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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