An inverse first exit problem for Wiener process is considered. This problem is to find such a ragion $G\colon(0,0)\in G\cup\partial G\subset R_+\times R$ that a distribution of the first exit point from it has a priory given property. For to estimate of this region with the uniform distribution of the first exit point a theorem on comparing of densities and a theorem on moment characterization are used. When the uniform density tends to zero and to infinity two asymptotics are investigated.