Random variable $X$ can take the following values: $-2,-1,0,1,2$. Find the probabilities of having such values if $E(X)=E(X^3) = 0$, $E(X^2) = 1$ and $E(X^4) = 2$.
$E(X) = -2P(X=-2)-1P(X=-1)+0P(X=0) + 1P(X=1) + 2P(X=2)=0$ $E(X^3) = -8P(X=-2)-1P(X=-1)+0P(X=0) + 1P(X=1) + 8P(X=2)=0$ $E(X^2) = 4P(X=-2)+1P(X=-1)+0P(X=0) + 1P(X=1) + 4P(X=2)=1$ $E(X^4) = 16P(X=-2)+1P(X=-1)+0P(X=0) + 1P(X=1) + 16P(X=2)=2$
Solving this equations I get $0.042, 0.333, 0, 0.042, 0.333$ What is wrong with the approach?