When can an ex post distribution be used to approximate an ex ante distribution?

An ex post distribution can effectively approximate an ex ante distribution under conditions where the distribution is stationary and the dataset is sufficiently large.

Stationarity implies that the statistical properties of a process generating the data (such as mean and variance) are constant over time. This stability ensures that the behavior observed in the historical data will likely hold in the future, making it a suitable basis for predictions and estimations. When the sample size is large enough, the Central Limit Theorem suggests that the sampling distribution of the mean will approach a normal distribution, regardless of the shape of the underlying population distribution. This further supports the reliability of the ex post distribution as a predictor of the ex ante distribution.

In contrast, options referencing minimal data would likely lead to unreliable estimates. A dataset that is normally distributed may not necessarily satisfy the conditions needed for a robust approximation, as the underlying characteristics and stability of the distribution also play crucial roles. Additionally, aligning with past performance without considering the conditions of stationarity and size may lead to misleading conclusions, as future performance can deviate from historical trends.