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Let (X1 , X2) be independent random variables. X1 has mean 0 and variance 1, while X2 has mean 1 and variance 4. The mutual information I (X1; X2) between X1 and X2 in bits is
Correct answer is '0.0 to 0.0'. Can you explain this answer?
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Verified Answer
Let (X1 , X2) be independent random variables. X1 has mean 0 and varia...
For two independent random variable
I(X;Y) = H(X) = H(X/Y)
H(X/Y) = H(X) for independent X and Y
⇒ I(X; Y) = 0
I(X;Y) = H(X) = H(X/Y)
H(X/Y) = H(X) for independent X and Y
⇒ I(X; Y) = 0
Most Upvoted Answer
Let (X1 , X2) be independent random variables. X1 has mean 0 and varia...
Calculating Mutual Information
The mutual information I(X1;X2) between X1 and X2 is given by:
I(X1;X2) = H(X1) + H(X2) - H(X1,X2)
Where H(X1) and H(X2) are the entropies of X1 and X2 respectively, and H(X1,X2) is the joint entropy of X1 and X2.
Calculating Entropy
The entropy of a random variable X is given by:
H(X) = -Σ p(x) log2 p(x)
Where p(x) is the probability mass function (PMF) of X.
Calculating Joint Entropy
The joint entropy of two random variables X1 and X2 is given by:
H(X1,X2) = -ΣΣ p(x1,x2) log2 p(x1,x2)
Where p(x1,x2) is the joint probability mass function (PMF) of X1 and X2.
Calculating Mutual Information for X1 and X2
Given that X1 and X2 are independent, we have:
p(x1,x2) = p(x1) p(x2)
Therefore, the joint entropy of X1 and X2 is:
H(X1,X2) = -ΣΣ p(x1) p(x2) log2 (p(x1) p(x2))
= -ΣΣ p(x1) p(x2) (log2 p(x1) + log2 p(x2))
= -Σ p(x1) log2 p(x1) Σ p(x2) log2 p(x2)
= H(X1) + H(X2)
Thus, the mutual information between X1 and X2 is:
I(X1;X2) = H(X1) + H(X2) - H(X1,X2)
= H(X1) + H(X2) - (H(X1) + H(X2))
= 0
Therefore, the correct answer is '0.0 to 0.0'.
The mutual information I(X1;X2) between X1 and X2 is given by:
I(X1;X2) = H(X1) + H(X2) - H(X1,X2)
Where H(X1) and H(X2) are the entropies of X1 and X2 respectively, and H(X1,X2) is the joint entropy of X1 and X2.
Calculating Entropy
The entropy of a random variable X is given by:
H(X) = -Σ p(x) log2 p(x)
Where p(x) is the probability mass function (PMF) of X.
Calculating Joint Entropy
The joint entropy of two random variables X1 and X2 is given by:
H(X1,X2) = -ΣΣ p(x1,x2) log2 p(x1,x2)
Where p(x1,x2) is the joint probability mass function (PMF) of X1 and X2.
Calculating Mutual Information for X1 and X2
Given that X1 and X2 are independent, we have:
p(x1,x2) = p(x1) p(x2)
Therefore, the joint entropy of X1 and X2 is:
H(X1,X2) = -ΣΣ p(x1) p(x2) log2 (p(x1) p(x2))
= -ΣΣ p(x1) p(x2) (log2 p(x1) + log2 p(x2))
= -Σ p(x1) log2 p(x1) Σ p(x2) log2 p(x2)
= H(X1) + H(X2)
Thus, the mutual information between X1 and X2 is:
I(X1;X2) = H(X1) + H(X2) - H(X1,X2)
= H(X1) + H(X2) - (H(X1) + H(X2))
= 0
Therefore, the correct answer is '0.0 to 0.0'.
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Let (X1 , X2) be independent random variables. X1 has mean 0 and variance 1, while X2 has mean 1 and variance 4. The mutual information I (X1; X2) between X1 and X2 in bits isCorrect answer is '0.0 to 0.0'. Can you explain this answer?
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