I have the following problem: Im looking for a probability distribution (i.e. a random variable with this distribution) that fullfills the following properties:
- $\mathbb{P}(X=0)=\delta$ where $\delta \in (0,1)$ can be chosen.
- $\mathbb{E}(X^2)=1$
- $\mathbb{E}(X^4)=1$
If such a distribution does not exist, I would be interested in a proof showing this.
So far i have tried to design a discrete distribution fullfilling the above requirements, but the corresponding linear system doesn't seem to be solvable or I end up with probabilities below zero or above one. Further one can design a distribution fullfilling the first and either the second or the third requirement, but it seems to be really hard to fullfill both the second and the third.
Thanks for your help, John