If you perform the following procedure:
1. Draw A from Poisson with mean x.
2. Sum up A i.i.d. +1/-1 Bernoulli variables with mean y.

You'll get a Gaussian random variable, with -- expectedly -- mean of x*y, but somewhat surprisingly with variance x.

It's somewhat nice (and surprising for me) that the variance is independent of the Bernoulli distribution skew.