  Consider a measure $\mu$ on $\R$ generating a natural exponential family
$F(\mu)$ with variance function $V_{F(\mu)}(m)$ and Laplace transform $$
e^{\ell_{\mu}(s)}=\int_{-\infty} ^{\infty}\exp(-sx)\mu(dx).$$ A dual measure
$\mu^*$ satisfies $\ell'_{\mu^*}(\ell'_{\mu}(s))=s.$ One important property is
$\ell''_{\mu^*}(m)=1/V_{F(\mu)}(m),$ leading to the notion of duality among
exponential families (or rather among the extended notion of T exponential
families $TF$ obtained by considering all translations of a given exponential
family $F$). For instance for Tweedie families of parameter $p$ and $q$ with
$\frac{1}{p}+\frac{1}{q}=1$ with $p>1$, then the corresponding T exponential
families are dual of each other. Duality is introduced by the need to
understand the strange formulas occuring in large deviations. The paper gives
details about two families: the one with variance function $e^m$ generated by
the unsymmetric stable measure with parameter one, also called Landau
distribution, which is dual of the Poisson distributions. This Landau
distribution is the building block for computing the duals of the Morris and
cubic exponential families. The second one has variance function $e^m-1$. It is
generated by a self dual probability on nonegative integers. It is called the
dilogarithm probability for its close ties with the dilogarithm function. We
give examples of measures which have no dual, and we compute a number of pairs
of families in duality.
