## 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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Page 75

Consider two random variables

one and only one ordered pair of numbers Xx (c) = x\ , X2(c) = x2. The space of

Consider two random variables

**Xx and X2**, which assign to each element c of ^one and only one ordered pair of numbers Xx (c) = x\ , X2(c) = x2. The space of

**Xx and X2**is the set of ordered pairs sd = {(x,, x2) : x, = Xx{c), x2 = X2(c), cec€}.Page 102

It will be shown that

density functions are /.(x.) = f(xux2)dx2 = (xx + x2) dx2 = x] + A, 0 < jc| < 1 , = 0

elsewhere, and Mx2) = f(xi,x2)dxi = (x, + x2) dxx = 5 + x2, 0 < x2 < 1, = 0

elsewhere.

It will be shown that

**Xx and X2**are dependent. Here the marginal probabilitydensity functions are /.(x.) = f(xux2)dx2 = (xx + x2) dx2 = x] + A, 0 < jc| < 1 , = 0

elsewhere, and Mx2) = f(xi,x2)dxi = (x, + x2) dxx = 5 + x2, 0 < x2 < 1, = 0

elsewhere.

Page 165

p.d.f. of the two new random variables Yx = ux(XuX2) and Y2 = u2(

given by g(yx » yi) = Rwx (yx , y2), w2(yx ,y2)], ( v, , y2) e <#, = 0 elsewhere,

where x, = WiO^, v2), x2 = w2(.y,,.y2) is the single-valued inverse of v, = «,(x,, x2),

...

p.d.f. of the two new random variables Yx = ux(XuX2) and Y2 = u2(

**Xx**,**X2**) isgiven by g(yx » yi) = Rwx (yx , y2), w2(yx ,y2)], ( v, , y2) e <#, = 0 elsewhere,

where x, = WiO^, v2), x2 = w2(.y,,.y2) is the single-valued inverse of v, = «,(x,, x2),

...

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Accordingly approximate best critical region 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 F-distribution gamma distribution given H0 is true 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 Yx percent confidence interval 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 statistic for 9 subset testing H0 theorem u(Xu X2 unbiased estimator XuX2 Xx and X2 Yu Y2 zero elsewhere