Under a standard hardness assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all polynomial-time samplable distributions. More precisely we show that if exponential time is not in subexponential space, then the following are equivalent for any algorithm $A$: for all $p$-samplable distributions $\mu$, $A$ runs in time polynomial on $\mu$-average; for all polynomial $p$, the running time for A is bounded by $2^{O(K^p(x)-K(x)+\log|x|)}$ for "all" inputs $x$. To prove this result we explore the time-bounded Kolmogorov distribution, $m^t(x)=2^{-K^t(x)}$ where $K^t(x)$ is the Kolmogorov complexity (smallest program size to generate $x$) with programs limited to run in time $t(|x|)$ and show that under the hardness assumption, the time-bounded Kolmogorov distribution is a universal samplable distribution.