## Introduction to Mathematical Statistics |

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Page 134

^*, 0 < x < oo, zero elsewhere. Show that X1 X1 + X2 = --> Y., - --→ * T X, + X, *

T X, + X, + X, Y Ya = X1 + X2 + Xa are mutually stochastically independent. 4.37.

**Let X1**, X2, Xa denote a random sample from the distribution having p.d. f. f(x) = e^*, 0 < x < oo, zero elsewhere. Show that X1 X1 + X2 = --> Y., - --→ * T X, + X, *

T X, + X, + X, Y Ya = X1 + X2 + Xa are mutually stochastically independent. 4.37.

Page 143

X2 and Z = XH + X3. Show that the momentgenerating function of the joint

distribution of Y and Z is exp [t?/(1 – 2t2)] E{exp [ti (X1 + X2) + ta(X} + X3)} = 1 –

2t2 for ...

**Let X1**, X2 be a random sample from the normal distribution n(0, 1). Let Y = X1 +X2 and Z = XH + X3. Show that the momentgenerating function of the joint

distribution of Y and Z is exp [t?/(1 – 2t2)] E{exp [ti (X1 + X2) + ta(X} + X3)} = 1 –

2t2 for ...

Page 267

Consider the simple hypothesis Ho: 6 = 0 = 2 and the alternative hypothesis H1:

0 = 0" = 4.

Show that the best test of Ho against H1 may be carried out by use of the statistic

XI ...

Consider the simple hypothesis Ho: 6 = 0 = 2 and the alternative hypothesis H1:

0 = 0" = 4.

**Let X1**, X2 denote a random sample of size 2 from this distribution.Show that the best test of Ho against H1 may be carried out by use of the statistic

XI ...

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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