The Exponential Distribution in Information Theory

It is shown that the exponential distribution leads to information theoretic formulas which are strikingly similar to their Gaussian counterparts:
$\bullet$ A saddle-point property satisfied by the mutual information between a random variable and its sum with an exponential random variable.
$\bullet$ Rate-distortion function of the Poisson process.
$\bullet$ Capacity of single-user and multiaccess channels with additive exponential noise.
$\bullet$ Capacity of Controlled Markov Processes.