## Introduction to Mathematical Statistics |

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Page 142

Whenever the limiting distribution of a random variable is a degenerate

distribution, the random variable is said to

that has a probability of one. Thus Examples 1 to 3 illustrate not only the notion of

a ...

Whenever the limiting distribution of a random variable is a degenerate

distribution, the random variable is said to

**converge stochastically**to the constantthat has a probability of one. Thus Examples 1 to 3 illustrate not only the notion of

a ...

Page 145

Find My(t; n), show that the o My(t; n) = e”, and deduce that Y = S^

1)]/V/2(n) — 1) = [ny – (n − 1)a?]/o”v/2(n) — 1)? 4. The Central Limit Theorem.

Find My(t; n), show that the o My(t; n) = e”, and deduce that Y = S^

**converges****stochastically**to go. What, however, would be the limiting distribution of [Z – (n −1)]/V/2(n) — 1) = [ny – (n − 1)a?]/o”v/2(n) — 1)? 4. The Central Limit Theorem.

Page 151

But e” is the moment-generating function of a degenerate distribution with

probability one at the point u. Thus the limiting distribution of X is degenerate (or

X

by ...

But e” is the moment-generating function of a degenerate distribution with

probability one at the point u. Thus the limiting distribution of X is degenerate (or

X

**converges stochastically**to a), and the limiting distribution function of X is givenby ...

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Accordingly best critical region binomial distribution cent confidence interval chi-square distribution composite hypothesis conditional p.d.f. confidence interval Consider continuous type degrees of freedom denote a random discrete type distribution function distribution having p.d.f. distribution with mean ExAMPLE Exercises f(zi function F(z hypothesis H1 independent random variables integral Jacobian joint p.d.f. joint sufficient statistics Let the random Let X1 limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent Mx(t My(t normal distribution n(z null hypothesis null simple hypothesis one-to-one transformation order statistics p.d.f. of Yi Poisson distribution positive integer Pr(a Pr(X Pr(Y probability density functions quadratic form random experiment random interval random sample random variables X1 respectively ſ ſ sample space Show significance level ſº stochastically independent random subset sufficient statistic Yi theorem tion type of random unbiased statistic values X1 and X2 Xn denote zero elsewhere