Visualize Cramér's theorem: the probability of a sample mean deviating from its expectation decays exponentially. The rate function I(x) = sup_λ{λx - log M(λ)} is the Legendre transform of the log-moment generating function.
Cramér's theorem (1938): P(S_n/n ≥ x) ≍ exp(-n·I(x)) where I(x) = sup_λ{λx - Λ(λ)} and Λ(λ)=log E[e^{λX}]. Gärtner-Ellis theorem generalizes. Entropy: I(p) = p·log(p/μ) + (1-p)·log((1-p)/(1-μ)) for Bernoulli (relative entropy / KL divergence). Sanov's theorem: probability of empirical measure Q given P is exp(-n·D_KL(Q||P)).