## Introduction to Mathematical Statistics |

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Page 75

If X1, X2, and Xa are

stochastically independent (that is, X, and X, i # j, where i, j = 1, 2, 3 are

stochastically independent). However, the following example, due to S. Bernstein

, shows that ...

If X1, X2, and Xa are

**mutually stochastically independent**, they are pairwisestochastically independent (that is, X, and X, i # j, where i, j = 1, 2, 3 are

stochastically independent). However, the following example, due to S. Bernstein

, shows that ...

Page 77

Let X1, X2, Xa, and X, be four

variables, each with p.d. f. f(r) = 3(1 — r)”, 0 < r < 1, zero elsewhere. If Y is the

minimum of these four variables, find the distribution function and the p.d.f. of Y.

2.32.

Let X1, X2, Xa, and X, be four

**mutually stochastically independent**randomvariables, each with p.d. f. f(r) = 3(1 — r)”, 0 < r < 1, zero elsewhere. If Y is the

minimum of these four variables, find the distribution function and the p.d.f. of Y.

2.32.

Page 134

Show that X1 X1 + X2 = --> Y., - --→ * T X, + X, * T X, + X, + X, Y Ya = X1 + X2 +

Xa are

1, ...

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,..., X, be r**mutually****stochastically independent**gamma variables with parameters a = 0, and 8 = 1, i =1, ...

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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