Q.3.26 Prove the equivalence of Equatio... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 3: Q.3.26 (page 109)

Prove the equivalence of Equations (5.11) and (5.12).

Short Answer

Expert verified

P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F) 鈬� P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)

Both directions are proven so the equivalence is correct.

Step by step solution

Given Information

Prove:

P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F) 鈬� P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)

Both of those are the defining conditions of conditional independence.

Explanation

Suppose that for $E_{1}, E_{2}, F$ the right equation ${}^{*}: P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)$ is true.

$P (E_{1} 鈭� E_{2} F) = \frac{P (E_{1} E_{2} F)}{P (E_{2} F)} = \frac{\frac{P (E_{1} E_{2} F)}{P (F)}}{\frac{P (E_{2} F)}{P (F)}} = \frac{P (E_{1} E_{2} 鈭� F)}{P (E_{2} 鈭� F)} \overset{*}{=} P (E_{1} 鈭� F)$

Taking the leftmost and the rightmost expression this proves the left equation in the hypothesis, i.e.:

P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F) 鈬� P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)

Explanation

For the other direction, suppose that for $E_{1}, E_{2}, F$ the right equation ${}^{* *}: P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F)$ is true.

$P (E_{1} E_{2} 鈭� F) = \frac{P (E_{1} E_{2} F)}{P (F)} = \frac{P (E_{1} 鈭� E_{2} F) P (E_{2} 鈭� F) P (F)}{P (F)} = P (E_{1} 鈭� E_{2} F) P (E_{2} 鈭� F)$

$\overset{* *}{=} P (E_{1} 鈭� F) P (E_{2} 鈭� F)$

This proves the right equation in the hypothesis, i.e.:

P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F) 鈬� P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)

Both directions are proven so the equivalence is correct.

Final Answer

$P (E_{1} 鈭� E_{2} F) = P (E_{1} 鈭� F) 鈬� P (E_{1} E_{2} 鈭� F) = P (E_{1} 鈭� F) P (E_{2} 鈭� F)$

Both directions are proven so the equivalence is correct.

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91影视

Prove the equivalence of Equations (5.11) and (5.12).

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

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