I Probability paradox: P(X=x)=1/n >1

AI Thread Summary
The discussion centers on a probability paradox involving a random variable X with a uniform distribution in the range (0,n), where n<1. It highlights the confusion between probability density functions and actual probabilities, clarifying that while P(X=x) can exceed 1/n, it does not violate probability laws because P(X=x) is always zero. The integral of the probability density function must remain at or below 1, which is satisfied in this case. An analogy with a uniform rod illustrates that density can be greater than the total mass without contradicting fundamental principles. The thread concludes with the resolution of the initial question and a reminder that probability density and probability are distinct concepts.
Trollfaz
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I have a random variable X in range(0,n) where n<1, with a uniform distribution
Then the probability of sample space S=n x P(X=x) x<=n which must be 1
Manipulating the equation P(X=x)=1/n >1
Then this violates the fundamental law of probability which says that any probability must be at most 1.
How do we resolve this paradox here
 
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Are you assuming that the maximum value a probability density function may take is ##1##?

That's certainly not the case.
 
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Trollfaz said:
Then this violates the fundamental law of probability which says that any probability must be at most 1.
You are confusing a probability density with a probability. The function you are describing is the probability density function and it can be arbitrarily large. The integral of a probability density is a probability. So only its integral must be no greater than 1, which is the case.
 
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Just to say it explicitly, P(X=x) is always zero. ##P(|X-x|<\epsilon)\approx 2\epsilon f(x)## is the right interpretation of the density function
 
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Trollfaz said:
I have a random variable X in range(0,n) where n<1, with a uniform distribution

If you had a uniform rod of mass 1 kg and length 1/2 meter, the density of the rod would be 2 kg per meter, even though the rod only has mass 1 kg. As the others pointed out, probability density is not the the same concept as probability.
 
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An off-topic discussion has been deleted, and since the OP's question has been answered, this thread is now closed. Thanks folks.
 
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