## Introduction to Mathematical Statistics |

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Page 46

EI(X — b)*] = 0. A discrete distribution of this extreme form is called a degenerate

distribution. 1.73. Let X be a random variable such that K(t) = E(t”) exists for all ...

**Prove**that the p.d.f. of X is of the form f(x) = 1, a = b, zero elsewhere, if and only ifEI(X — b)*] = 0. A discrete distribution of this extreme form is called a degenerate

distribution. 1.73. Let X be a random variable such that K(t) = E(t”) exists for all ...

Page 352

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.

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

A. Hint.

Page 358

Let A1, A2, ..., Ao be the matrices of k > 2 quadratic forms Qı, Q2, ..., Q, in the

items of a random sample of size n from a distribution which is n(0, 0°).

the pairwise stochastic independence of these forms implies that they are

mutually ...

Let A1, A2, ..., Ao be the matrices of k > 2 quadratic forms Qı, Q2, ..., Q, in the

items of a random sample of size n from a distribution which is n(0, 0°).

**Prove**thatthe pairwise stochastic independence of these forms implies that they are

mutually ...

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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