  In this paper, we consider a fractional Poisson random field (FPRF) on
positive plane. It is defined as a process whose one dimensional distribution
is the solution of a system of fractional partial differential equations. A
time-changed representation for the FPRF is given in terms of the composition
of Poisson random field with a bivariate random process. Some integrals of the
FPRF are introduced and studied. Using the Adomian decomposition method, a
closed form expression for its probability mass function is obtained in terms
of the generalized Wright function. Some results related to the order
statistics of random numbers of random variables are presented. Also, we
introduce a generalization of Poisson random field on $\mathbb{R}^d_+$, $d\ge1$
which reduces to the Poisson random field in a special case. For $d=1$, it
further reduces to a generalized Poisson process (GPP). A time-changed
representation for the GPP is established. Moreover, we construct a
time-changed linear and planer random motions of a particle driven by the GPP.
The conditional distribution of the random position of particle is derived.
Later, we define the compound fractional Poisson random field via FPRF. Also, a
generalized version of it on $\mathbb{R}^d_+$, $d\ge1$ is discussed.
