## Introduction to Mathematical StatisticsAn exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform. |

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Expectations of Functions of One Random Variable /•oo E[u(X)] = Ydu(x)f(x) or u(

x)f(x)dx "-oo p = E(X) and a2 = E[(X - pi)2] = EiX2) - n2 M(t) = E(e'x) and \i = M' (0),

a2 = M"(0) - [AT (0)]2 Multivariate Distributions f(

Expectations of Functions of One Random Variable /•oo E[u(X)] = Ydu(x)f(x) or u(

x)f(x)dx "-oo p = E(X) and a2 = E[(X - pi)2] = EiX2) - n2 M(t) = E(e'x) and \i = M' (0),

a2 = M"(0) - [AT (0)]2 Multivariate Distributions f(

**xux2**)dx2 x2 J_ f2\l(x2\xi) — f(xi ...Page 192

Let

zero elsewhere. Show that Y,=X^T2' Y2 = Xl + X2 + Xi' Yj-Xx + X* + Xj are

mutually independent. 4.50. Let Xu X2, . . . , Xr be r independent gamma

variables with ...

Let

**XuX2**,X3 be i.i.d., each with the distribution having p.d.f. f(x) = e~x, 0 < x < oo,zero elsewhere. Show that Y,=X^T2' Y2 = Xl + X2 + Xi' Yj-Xx + X* + Xj are

mutually independent. 4.50. Let Xu X2, . . . , Xr be r independent gamma

variables with ...

Page 325

To prove this, we write, in accordance with the factorization theorem, f(X\ ; 6) □ . 'f

(x„; 6) = k\[ux(xu x2, . . . , x„); 9]k2(xu x2, . . . , x„). However, we find that j>, = w(z)

or, equivalently, ux(

...

To prove this, we write, in accordance with the factorization theorem, f(X\ ; 6) □ . 'f

(x„; 6) = k\[ux(xu x2, . . . , x„); 9]k2(xu x2, . . . , x„). However, we find that j>, = w(z)

or, equivalently, ux(

**xux2**, . . . , x„) = w[v(xu x2, . . . , x„)], which is not a function of 9...

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Accordingly approximate best critical region bivariate normal distribution chi-square distribution complete sufficient statistic conditional p.d.f. conditional probability confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random depend upon 9 discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters equation estimator of 9 Example Exercise gamma distribution given H0 is true hypothesis H0 independent random variables integral joint p.d.f. Let the random Let Xu X2 likelihood function limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Xu Poisson distribution positive integer probability density functions probability set function quadratic form random experiment random sample random variables Xx reject H0 respectively sample space Section Show significance level simple hypothesis statistic for 9 testing H0 theorem u(Xu X2 unbiased estimator variance a2 XuX2 Xx and X2 Yu Y2 zero elsewhere