Probability paradox: P(X=x)=1/n >1

Trollfaz · Nov 19, 2022

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

PeroK · Nov 19, 2022

Are you assuming that the maximum value a probability density function may take is ##1##?

That's certainly not the case.

Dale · Nov 19, 2022

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.

Office_Shredder · Nov 19, 2022

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

Stephen Tashi · Nov 20, 2022

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.

berkeman · Nov 1, 2023

An off-topic discussion has been deleted, and since the OP's question has been answered, this thread is now closed. Thanks folks.

Probability paradox: P(X=x)=1/n >1

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