## Introduction to Mathematical Statistics |

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Page 194

It is easy to show (

– 1) degrees of freedom. Since Q. = ax (X. — X)' > 0, the theorem enables i-l s

that ically nt and that Q4/o” has a chisquare distribution with ab – 1 – boa - 1) = b

- 1 ...

It is easy to show (

**Exercise**10.1) that Q:/g” has a chi-square distribution with b(a– 1) degrees of freedom. Since Q. = ax (X. — X)' > 0, the theorem enables i-l s

that ically nt and that Q4/o” has a chisquare distribution with ab – 1 – boa - 1) = b

- 1 ...

Page 205

8* = 0, or Ho: us = u + x., XXx: = 0, against the alternative composite hypothe1 d b

sis Hi: us, = u + x; + 8, XXx: = XX3; - 0, may be based on an F 1 1 statistic. The

constant c is so selected as to yield the desired value of oz.

8* = 0, or Ho: us = u + x., XXx: = 0, against the alternative composite hypothe1 d b

sis Hi: us, = u + x; + 8, XXx: = XX3; - 0, may be based on an F 1 1 statistic. The

constant c is so selected as to yield the desired value of oz.

**Exercises**• 10.7.Page 220

However the converse is not in general true as was seen in

and 11.10, page 216. The importance of the theorem lies in the fact that we now

know when, and only when, two random variables having a bivariate normal ...

However the converse is not in general true as was seen in

**Exercises**11.1(c)and 11.10, page 216. The importance of the theorem lies in the fact that we now

know when, and only when, two random variables having a bivariate normal ...

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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