 Exponential tilting

exponential tilting reweights a probability law in favour of larger or smaller values. Let $K(\theta )=\log {\mathbb{E}}_P{e}^{\theta X}$ be finite at the chosen parameter. Define the tilted law by its Radon-Nikodym derivativeIts total mass is one by construction. Positive $\theta$ favours larger $x$, while negative $\theta$ favours smaller $x$; it changes probabilities without creating values outside the original support.

For $\theta$ in the interior of the finite moment-generating-function domain, differentiation under the integral givesThe interior-domain condition supplies the integrable exponential bounds needed for these derivatives. Endpoint tilts can exist without a finite tilted mean, so existence of a normalized tilt alone is insufficient for the mean formula.

For a Bernoulli distribution with success probability $p\in (0,1)$, the tilted success probability is $p_{\theta }=p{e}^{\theta }/(1-p+p{e}^{\theta })$: tilting adds $\theta$ to its log odds. A Normal distribution $N(\mu ,{\sigma }^2)$ tilts to $N(\mu +{\sigma }^2\theta ,{\sigma }^2)$ by completing the square. These calculations explain why exponential tilting is useful for making a rare Sample mean typical while retaining an explicit Likelihood ratio.