  We construct the quasi probability distribution $W(p,q)$ on even dimensional
vector space with marginality and invariance under the transformation induced
by projective representation of the group ${\rm Sp}(2,\mathbb{Z})$ whose
elements correspond to linear canonical transformation.
  On even dimensional vector space, non-existence of such a quasi probability
distribution whose arguments take physical values was shown in our previous
paper(Phys.Rev.A{\bf 65} 032105(2002)). For this reason we study a quasi
probability distribution $W(p,q)$ whose arguments $q$ and $p$ take not only $N$
physical values but also $N$ unphysical values, where $N$ is dimension of
vector space. It is shown that there are two quasi probability distributions on
even dimensional vector space. The one is equivalent to the Wigner function
proposed by Leonhardt, and the other is a new one.
