what does it mean when $\operatorname{E}[X^2]$ diverges? is it possible for a random variable $X$, such that the expected value of $X^2$, $\operatorname{E}[X^2]$ is a divergent integral?
If it is impossible, does that mean the probability density function of $X$ is wrong? (the integral of the probability density function over the support does not equal to $1$)
Also, since $\operatorname{Var}[X] = \operatorname{E}[X^2] - \operatorname{E}[X] ^ 2$, does $\operatorname{Var}[X]$ still exist in this case?

This simply mean that the variance does not exist for this random variable. Nothing more! The probability density function can be perfectly correct.
There are several example for this kind of random variables:


*

*Some Student distributions

*Cauchy random variables

*...

