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Published in:
1977 | OriginalPaper | Chapter
Sampling Distributions
Authors : A. M. Mathai, P. N. Rathie
Published in: Probability and Statistics
Publisher: Palgrave Macmillan UK
Included in: Professional Book Archive
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In Chapter 2 we defined the independence of two events A and B. That is, A and B are said to be independent if, <math display='block'> <mrow> <mi>P</mi><mrow><mo>(</mo> <mrow> <mi>A</mi><mo>∩</mo><mi>B</mi> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow><mi>P</mi><mrow><mo>(</mo> <mi>B</mi> <mo>)</mo></mrow> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\[P\left( {A \cap B} \right) = P\left( A \right)P\left( B \right)\]$$ Consider two stochastic variables X and r having a joint distribution and let the events A and B be, <math display='block'> <mrow> <mi>A</mi><mo>=</mo><mrow><mo>{</mo> <mrow> <mi>x</mi><mtext> </mtext><mrow><mo>|</mo><mrow> <mtext> </mtext><mi>a</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>x</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>b</mi> </mrow></mrow> </mrow> <mo>}</mo></mrow><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>B</mi><mtext> </mtext><mo>=</mo><mrow><mo>{</mo> <mrow> <mi>y</mi><mrow><mo>|</mo><mrow> <mtext> </mtext><mi>c</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>y</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>d</mi> </mrow></mrow> </mrow> <mo>}</mo></mrow> </mrow> </math> ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\[A = \left\{ {x\;\left| {\;a\;\angle \;x\;\angle \;b} \right.} \right\}\;and\;B\; = \left\{ {y\left| {\;c\;\angle \;y\;\angle \;d} \right.} \right\}\]$$ , then, is shown in Figure 7.1. <math display='block'> <mrow> <mi>A</mi><mtext> </mtext><mo>∩</mo><mtext> </mtext><mi>B</mi><mtext> </mtext><mo>=</mo><mrow><mo>{</mo> <mrow> <mrow><mo>(</mo> <mrow> <mi>x</mi><mo>,</mo><mi>y</mi> </mrow> <mo>)</mo></mrow><mtext> </mtext><mrow><mo>|</mo><mrow> <mtext> </mtext><mi>a</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>x</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>b</mi><mo>,</mo><mtext> </mtext><mi>c</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>y</mi><mtext> </mtext><mo>∠</mo><mtext> </mtext><mi>d</mi> </mrow></mrow> </mrow> <mo>}</mo></mrow> </mrow> </math> ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\[A\; \cap \;B\; = \left\{ {\left( {x,y} \right)\;\left| {\;a\;\angle \;x\;\angle \;b,\;c\;\angle \;y\;\angle \;d} \right.} \right\}\]$$