Chapter 3: Q.3.28 (page 109)

Prove or give a counterexample. If $E_{1}$ and $E_{2}$ are independent, then they are conditionally independent given $F$ .

Short Answer

Expert verified

Rolling two independent six sided dice.

$E_{1}$ - the number on the first die is $1$ , $E_{2}$ - the number on the second die is $4$ , $F$ - the sum of the numbers is on both dice is $5$ .

Step by step solution

Given Information

Independent events $E_{1}, E_{2}$ .

Event $F$ .

Explanation

Do $E_{1}$ and $E_{2}$ have to be conditionally independent given $F$ .

Not true

Counterexample: Rolling two independent six sided dice. $E_{1}$ - the number on the first die is $1$ , $E_{2}$ - the number on the second die is $4$ , $F$ - the sum of the numbers is on both dice is 5 .

$P (E_{1}) = \frac{1}{6}$

$P (E_{2}) = \frac{1}{6}$

$P (E_{1} E_{2}) = \frac{1}{36}$

$P (E_{2} E_{1}) = P (E_{1}) P (E_{2})$

Explanation

$P (F) = \frac{4}{36} = \frac{1}{9}$

P (E_{1} ∣ F) = \frac{1}{4}

$P (E_{2} ∣ E_{1} F) = 1$

$(E_{2} ∣ E_{1} F) \neq P (E_{1} ∣ F)$

Since the equality $P (E_{2} ∣ E_{1} F) \neq P (E_{1} ∣ F)$ is the definition of conditional independence these events are not conditionally independent.

Final Answer

Rolling two independent six sided dice.

$E_{1}$ - the number on the first die is $1$ , $E_{2}$ - the number on the second die is $4$ , $F$ - the sum of the numbers is on both dice is $5$ .

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Prove or give a counterexample. If $E_{1}$ and $E_{2}$ are independent, then they are conditionally independent given $F$ .

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

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Prove or give a counterexample. If E1 and E2 are independent, then they are conditionally independent given F.

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Logic and Functions

Probability and Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Prove or give a counterexample. If $E_{1}$ and $E_{2}$ are independent, then they are conditionally independent given $F$ .