## Introduction to Mathematical Statistics |

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Page 112

4.2

finding the distribution of a function of one or more random variables is called the

change of variable technique. There are some delicate questions (with particular

...

4.2

**Transformations**of Variables of the Discrete Type An alternative method offinding the distribution of a function of one or more random variables is called the

change of variable technique. There are some delicate questions (with particular

...

Page 117

Consider the random variable Y = u(X), where y = u(x) defines a one-to-one

denoted by a = w(y), and let the derivative da/dy = w'(y) be continuous and not

vanish ...

Consider the random variable Y = u(X), where y = u(x) defines a one-to-one

**transformation**which maps the set of onto the set 3. Let the inverse of y = u(x) bedenoted by a = w(y), and let the derivative da/dy = w'(y) be continuous and not

vanish ...

Page 118

The space & is & = {z; 0 < r < 1}, which the one-to-one

maps onto 3 = {y; 0 < y < oo}. The Jacobian of the

the p.d.f. g(y) of Y = –2 ln X is g(y) = f(e-woo)|J| = }e-w!”, 0 < y < oo, = 0 elsewhere,

...

The space & is & = {z; 0 < r < 1}, which the one-to-one

**transformation**y = -2 ln amaps onto 3 = {y; 0 < y < oo}. The Jacobian of the

**transformation**is Accordingly,the p.d.f. g(y) of Y = –2 ln X is g(y) = f(e-woo)|J| = }e-w!”, 0 < y < oo, = 0 elsewhere,

...

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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