## Introduction to Mathematical Statistics |

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Page 219

and the variance of p(Yi). 8.15. Let the random variables X and Y have the joint

p.d. f. f(x, y) = (2/6°)e^***, 0 < x < y < oo, zero elsewhere. (a)

...

**Show**that Ya is an unbiased statistic for 6 with variance 6°. Find E(Y|2|y) = p(yi)and the variance of p(Yi). 8.15. Let the random variables X and Y have the joint

p.d. f. f(x, y) = (2/6°)e^***, 0 < x < y < oo, zero elsewhere. (a)

**Show**that the mean...

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12.2. In Example 3, verify that Q = Q2 + (), + Q3. 12.3. Let X1, X2,..., Xn be a

random sample from a normal distribution n(u, g”).

where X = § X/n and X' = $ X/(n – 1). Hint. Replace X – X by i = 1 i = 2 (X, -X) = (X,

-X)/n.

12.2. In Example 3, verify that Q = Q2 + (), + Q3. 12.3. Let X1, X2,..., Xn be a

random sample from a normal distribution n(u, g”).

**Show**that X (X, -X-X, X-xy +where X = § X/n and X' = $ X/(n – 1). Hint. Replace X – X by i = 1 i = 2 (X, -X) = (X,

-X)/n.

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generating function of Q/g”. 13.7. Let A be a real symmetric matrix. Prove that

each of the nonzero characteristic numbers of A is equal to one if and only if A* =

A. Hint.

**Show**that Q/o” does not have a chi-square distribution. Find the moment-generating function of Q/g”. 13.7. Let A be a real symmetric matrix. Prove that

each of the nonzero characteristic numbers of A is equal to one if and only if A* =

A. Hint.

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accept accordance Accordingly alternative approximately assume called cent Chapter complete compute conditional confidence interval Consider constant continuous type critical region decision defined definition degrees of freedom denote a random depend determine discrete type distribution function equal Equation event Example EXERCISES exists expected fact given Hence inequality integral interval joint p.d.f. Let X1 likelihood marginal matrix maximum mean moment-generating function mutually stochastically independent normal distribution Note observed order statistics outcome parameter Pr(X probability density functions problem proof prove random experiment random interval random sample random variable ratio reject respectively result sample space Show significance level simple hypothesis ſº stochastically independent sufficient statistic symmetric matrix Table theorem transformation true unknown variance write X1 and X2 zero elsewhere