## Introduction to mathematical statistics |

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Page 77

2.30. lif(x1,x2) = e~xi~x2,0 < x1 < oo, 0 < x2 < oo, zero elsewhere, is the joint

p.d.f. of the

four ...

2.30. lif(x1,x2) = e~xi~x2,0 < x1 < oo, 0 < x2 < oo, zero elsewhere, is the joint

p.d.f. of the

**random variables**X1 and X2, show that X1 and X2 are stochastically**independent**and that £(e*Ği+*a>) = (1 - t)-*, t < 1. 2.31. Let -X-2, X3, and X4 befour ...

Page 142

Let X1 and X2 be stochastically

X1 + X2 have chi-square distributions with r1 and r degrees of freedom

respectively. Here r1 < r. Show that X2 has a chi-square distribution with r —

degrees of ...

Let X1 and X2 be stochastically

**independent random variables**. Let X1 and Y =X1 + X2 have chi-square distributions with r1 and r degrees of freedom

respectively. Here r1 < r. Show that X2 has a chi-square distribution with r —

degrees of ...

Page 149

Let X and Y be random variables with p1 = 1, p2 = 4, a? = 4, o| = 6, p = ^. Find the

mean and variance of Z = 3X — 2Y. 4.72. Let X and Y be stochastically

Determine the ...

Let X and Y be random variables with p1 = 1, p2 = 4, a? = 4, o| = 6, p = ^. Find the

mean and variance of Z = 3X — 2Y. 4.72. Let X and Y be stochastically

**independent random variables**with means fi1, p2 and variances a2, a$.Determine the ...

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Accordingly best critical region binomial distribution cent confidence interval Chapter chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges stochastically critical region decision function defined degrees of freedom denote a random discrete type distribution having p.d.f. Equation Example EXERCISES F distribution function of Y1 given H0 is true independent random variables inequality integral joint p.d.f. Let the random Let X1 limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral order statistics Poisson distribution positive integer power function Pr X1 probability density functions probability set function quadratic form random experiment random interval random sample random variables X1 reject H0 respectively sample space Show significance level simple hypothesis H0 statistic for 9 statistic Y1 stochastically independent random subset testing H0 theorem type of random unbiased statistic variance a2 X1 and X2 Xn denote zero elsewhere