Summary: Let \(\mu\) be a log-concave probability measure on \({\mathbb{R}}^n\) and for any \(N> n\) consider the random polytope \(K_N=\operatorname{conv}\{X_1,\dots ,X_N\} \), where \(X_1,X_2,\dots\) are independent random points in \({\mathbb{R}}^n\) distributed according to \(\mu \). We study the question if there exists a threshold for the expected measure of \(K_N\). Our approach is based on the Cramer transform \(\Lambda_{\mu}^*\) of \(\mu \). We examine the existence of moments of all orders for \(\Lambda_{\mu}^*\) and establish, under some conditions, a sharp threshold for the expectation \({\mathbb{E}}_{\mu^N}[\mu (K_N)]\) of the measure of \(K_N\): it is close to 0 if \(\ln N\ll{\mathbb{E}}_{\mu}(\Lambda_{\mu}^*)\) and close to 1 if \(\ln N\gg{\mathbb{E}}_{\mu}(\Lambda_{\mu}^*)\). The main condition is that the parameter \(\beta (\mu )=\mathrm{Var}_{\mu}(\Lambda_{\mu}^*)/({\mathbb{E}}_{\mu}(\Lambda_{\mu}^*))^2\) should be small.