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Let (X1, X2) be independent random varibales. 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'. Can you explain this answer?
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Most Upvoted Answer
Let (X1, X2) be independent random varibales. X1has mean 0 and varianc...
Understanding Mutual Information
Mutual information quantifies the amount of information obtained about one random variable through another. It is defined mathematically as:
- I(X1; X2) = H(X1) + H(X2) - H(X1, X2)
Where H denotes entropy.
Independence of Random Variables
In the given scenario, X1 and X2 are independent random variables. This independence implies:
- The occurrence of X1 provides no information about X2 and vice versa.
Implications of Independence
For independent random variables:
- I(X1; X2) = 0
This is because when two variables are independent, their joint entropy equals the sum of their individual entropies:
- H(X1, X2) = H(X1) + H(X2)
As a result, the mutual information, which measures shared information, becomes zero.
Specifics of X1 and X2
- X1 has a mean of 0 and variance of 1.
- X2 has a mean of 1 and variance of 4.
These parameters do not affect the independence property. Hence, while their distributions differ, they do not share any information.
Conclusion
Given that X1 and X2 are independent, the mutual information I(X1; X2) is:
- 0 bits
This reflects the absence of any predictive relationship between the two variables.
Mutual information quantifies the amount of information obtained about one random variable through another. It is defined mathematically as:
- I(X1; X2) = H(X1) + H(X2) - H(X1, X2)
Where H denotes entropy.
Independence of Random Variables
In the given scenario, X1 and X2 are independent random variables. This independence implies:
- The occurrence of X1 provides no information about X2 and vice versa.
Implications of Independence
For independent random variables:
- I(X1; X2) = 0
This is because when two variables are independent, their joint entropy equals the sum of their individual entropies:
- H(X1, X2) = H(X1) + H(X2)
As a result, the mutual information, which measures shared information, becomes zero.
Specifics of X1 and X2
- X1 has a mean of 0 and variance of 1.
- X2 has a mean of 1 and variance of 4.
These parameters do not affect the independence property. Hence, while their distributions differ, they do not share any information.
Conclusion
Given that X1 and X2 are independent, the mutual information I(X1; X2) is:
- 0 bits
This reflects the absence of any predictive relationship between the two variables.
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Community Answer
Let (X1, X2) be independent random varibales. X1has mean 0 and varianc...
Mutual information of two random variables is a measure to tell how much one random variable tells about the other.
It is mathematically defined as:
I(X1, X2) = H(X1) – H(X1/X2)
Application:
Since X1 and X2 are independent, we can write:
H(X1/X2) = H(X1)
I(X1,X2 ) = H(X1) – H(X1)
= 0
It is mathematically defined as:
I(X1, X2) = H(X1) – H(X1/X2)
Application:
Since X1 and X2 are independent, we can write:
H(X1/X2) = H(X1)
I(X1,X2 ) = H(X1) – H(X1)
= 0
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Let (X1, X2) be independent random varibales. X1has mean 0 and variance 1, while X2has mean 1 and variance 4. The mutual information I(X1; X2) between X1and X2in bits is_______.Correct answer is '0'. Can you explain this answer?
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