Let’s assume that a cumulative probability distribution F(x) exists.
Then the median must exist, It’s simply the value of x for which F(x)=0.5.

The standard deviation doesn’t have to exist.
For example the cumulative probability distribution F(x) = arctan(x) / pi + 0.5 doesn’t have a standard deviation. The standard deviation is infinite.

My question is this. Is there any cumulative probability distribution that doesn’t have a mean. … I think I already have an answer.

The cumulative probability distribution:
F(x) = 0 for x <= 0
F(x) = arctan(x) / (pi/2) for x > 0
doesn’t have a mean, right?

indeed, it is quite fitting that a thread “Is there such a thing as luck?” is followed by “Another probability question”

I agree that the given function has no mean.

The mean would be defined by the integral of xF divided by the integral of F.

dv said:

I agree that the given function has no mean.

The mean would be defined by the integral of xF divided by the integral of F.

Thank you, dv.

I did a simulation on Excel and found that for a finite number of samples, the sample mean increases roughly as the logarithm of the number of samples.

That makes sense as the integral of x times the pdf would be the integral of x/(1+x^2) which trends to the integral as 1/x for large x.