  Let $\mu_{g}$ and $\mu_{p}$ denote the Gaussian and Poisson measures on
${\Bbb R}$, respectively. We show that there exists a unique measure
$\widetilde{\mu}_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann
transform $S_{\mu_g}$ the space $L^2({\Bbb R},\mu_g)$ is isomorphic to the
space ${\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})$ of analytic $L^2$-functions
on ${\Bbb C}$ with respect to $\widetilde{\mu}_{g}$. We also introduce the
Segal-Bargmann transform $S_{\mu_p}$ for the Poisson measure $\mu_{p}$ and
prove the corresponding result. As a consequence, when $\mu_{g}$ and $\mu_{p}$
have the same variance, $L^2({\Bbb R},\mu_g)$ and $L^2({\Bbb R},\mu_p)$ are
isomorphic to the same space ${\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})$ under
the $S_{\mu_g}$ and $S_{\mu_p}$-transforms, respectively. However, we show that
the multiplication operators by $x$ on $L^2({\Bbb R}, \mu_g)$ and on $L^2({\Bbb
R}, \mu_p)$ act quite differently on ${\cal H}L^2({\Bbb C},
\widetilde{\mu}_{g})$.
