## Introduction to Mathematical Statistics |

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Page 46

Robert V. Hogg, Allen Thornton Craig ..., Xn are said to be mutually

from this definition of the mutual

Robert V. Hogg, Allen Thornton Craig ..., Xn are said to be mutually

**stochastically****independent**if, and only if, f(zi, a 2, ..., zn) = f(z)f(x2) ... f.(c.). It follows immediatelyfrom this definition of the mutual

**stochastic independence**of X1, X2, ..., Xn that ...Page 53

Let Xi and X, be

X2 have chi-square distributions with ri and r degrees of freedom, respectively.

Here ri < r. Show that X2 has a chi-square distribution with r — ri degrees of ...

Let Xi and X, be

**stochastically independent random**variables. Let X, and Y = XI +X2 have chi-square distributions with ri and r degrees of freedom, respectively.

Here ri < r. Show that X2 has a chi-square distribution with r — ri degrees of ...

Page 220

+ —g- - In Section 11.3 there will be investigated a test of the hypothesis that two

and Y ...

+ —g- - In Section 11.3 there will be investigated a test of the hypothesis that two

**random**variables having a bivariate normal distribution are**stochastically****independent**. The following theorem is essential to that test. THEOREM. Let Xand Y ...

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### Common terms and phrases

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