• I
• Trollfaz
In summary, the conversation discusses the concept of probability density and how it differs from probability. It is explained that the probability density function can be arbitrarily large, but the integral of the function must be no greater than 1. This leads to a paradox in the given scenario, but it is resolved by understanding the difference between probability density and probability.

#### Trollfaz

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

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

That's certainly not the case.

FactChecker
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.

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

Dale
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.

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

WWGD