## Introduction to Mathematical Statistics |

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Page 143

Let X1, X2 be a random sample from the

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 348

Let X1, X2,.. .., X, have a multivariate

the means and V is the positive definite covariance matrix. Let Y = c'X and Z = d'X

, where X = [X1, ..., Xn], c' = [c1, ..., ca), and d' = [di,..., d.] are real matrices.

Let X1, X2,.. .., X, have a multivariate

**normal distribution**, where p is the matrix ofthe means and V is the positive definite covariance matrix. Let Y = c'X and Z = d'X

, where X = [X1, ..., Xn], c' = [c1, ..., ca), and d' = [di,..., d.] are real matrices.

Page 381

Jacobian, 117, 118, 129, 132 Joint distribution function, 54 Joint probability

density function, 54 Law of large numbers, ... 142 of binomial distribution, 80 of

bivariate

distribution, ...

Jacobian, 117, 118, 129, 132 Joint distribution function, 54 Joint probability

density function, 54 Law of large numbers, ... 142 of binomial distribution, 80 of

bivariate

**normal distribution**, 105 of chi-square distribution, 93 of gammadistribution, ...

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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