## Introduction to Mathematical Statistics |

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Page 67

Henceforth, we shall refer to dr/dy = w'(y) as the

transformation. In most mathematical areas, J = w'(y) is referred to as the

the ...

Henceforth, we shall refer to dr/dy = w'(y) as the

**Jacobian**(denoted by J) of thetransformation. In most mathematical areas, J = w'(y) is referred to as the

**Jacobian**of the inverse transformation a = w(y), but in this book it will be calledthe ...

Page 68

continuous and that the

illustrative example may be desirable before we proceed with the extension of

the change of variable technique to two random variables of the continuous type.

continuous and that the

**Jacobian**J is not identically equal to zero in B. Anillustrative example may be desirable before we proceed with the extension of

the change of variable technique to two random variables of the continuous type.

Page 102

accordingly not in the

upon 0 and that Yı is a sufficient statistic for 6./ The converse is proved by taking g

(y1, y2, ..., ya; 0) = g1(y1; 0)h(y2, ..., ynly), where h(y2, ..., yayi) does not depend ...

accordingly not in the

**Jacobian**J, it follows that h(y2, ..., yayi; 6) does not dependupon 0 and that Yı is a sufficient statistic for 6./ The converse is proved by taking g

(y1, y2, ..., ya; 0) = g1(y1; 0)h(y2, ..., ynly), where h(y2, ..., yayi) does not depend ...

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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