## Introduction to Mathematical Statistics |

### From inside the book

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Page 8

then for every

A2) = 1 — p, P(A3) = p, P(A) = 1. Not all random experiments are sufficiently

simple to have the outcome denoted by a single symbol X. Suppose, for example,

that ...

then for every

**subset**A of A the set function P(A) = X's(a) is such that P(A) = 0, P(A2) = 1 — p, P(A3) = p, P(A) = 1. Not all random experiments are sufficiently

simple to have the outcome denoted by a single symbol X. Suppose, for example,

that ...

Page 9

for all

{(x, y); 0 < r < 1, 0 < y < 1}, ... Let there be a type of

such that a set function P(A) is defined for each A of that type. Henceforth all sets

...

for all

**subsets**A of A = {(x, y); 0 < z, 0 < y} for which the integral exists. Thus, if A ={(x, y); 0 < r < 1, 0 < y < 1}, ... Let there be a type of

**subset**of A, including A itself,such that a set function P(A) is defined for each A of that type. Henceforth all sets

...

Page 78

This theorem has a natural extension to n-fold integrals. This extension is as

follows: Consider an integral of the form J. Joe, 22, ''', wn) dai dao ... dao, A taken

over a

21, ...

This theorem has a natural extension to n-fold integrals. This extension is as

follows: Consider an integral of the form J. Joe, 22, ''', wn) dai dao ... dao, A taken

over a

**subset**A of an n dimensional space A. Let y1 = u1(zi, 22, ''', 2n), 92 = w2(21, ...

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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