## Introduction to mathematical statistics |

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Page 190

7.2 Stochastic Convergence When the limiting distribution of a random variable is

degenerate, the random variable is said to

constant that has a probability of one. Thus Examples 1 to 3 of Section 7.1

illustrate not ...

7.2 Stochastic Convergence When the limiting distribution of a random variable is

degenerate, the random variable is said to

**converge stochastically**to theconstant that has a probability of one. Thus Examples 1 to 3 of Section 7.1

illustrate not ...

Page 200

Theorem 4. Let Fn(u) denote the distribution function of a random variable U

whose distribution depends upon the positive integer n. Let U

Theorem 4. Let Fn(u) denote the distribution function of a random variable U

whose distribution depends upon the positive integer n. Let U

**converge****stochastically**to the constant c ^ 0. The random variable U /c**converges****stochastically**to one.Page 202

(Y/n)(l — Y/n)/[p(l — p)]

that the following does also: \(Y/n)(l - Y/n)y* [ P(l-P) J Thus, in accordance with

Theorem 6, the ratio W = U/V, namely, Y - np V»(Y/n)(1 - Y/n) has a limiting ...

(Y/n)(l — Y/n)/[p(l — p)]

**converges stochastically**to one, and Theorem 5 assertsthat the following does also: \(Y/n)(l - Y/n)y* [ P(l-P) J Thus, in accordance with

Theorem 6, the ratio W = U/V, namely, Y - np V»(Y/n)(1 - Y/n) has a limiting ...

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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