  With any symmetric distribution $\mu$ on the real line we may associate a
parametric family of noncentral distributions as the distributions of
$(X+\delta)^2$, $\delta\not=0$, where $X$ is a random variable with
distribution $\mu$.
  The classical case arises if $\mu$ is the standard normal distribution,
leading to the noncentral chi-squared distributions. It is well-known that
these may be written as Poisson mixtures of the central chi-squared
distributions with odd degrees of freedom. We obtain such mixture
representations for the logistic distribution and for the hyperbolic secant
distribution. We also derive alternative representations for chi-squared
distributions and relate these to representations of the Poisson family.
  While such questions originated in parametric statistics they also appear in
the context of the generalized second Ray-Knight theorem, which connects
Gaussian processes and local times of Markov processes.
