Simulations

Introduction Suppose data consisting of i.i.d. observations \(X^n=(X_1,X_2,\cdots,X_n)\) are available from a distribution \(F(x,\theta),\,\theta\in\Theta\subset\mathbf{R}.\) The exact value \(\theta\) corresponding to the distribution that generated the observations is unknown. The problem is, using the available data \(X^n,\) construct tests for making decisions on the possible value of unknown parameter \(\theta\).

Continuous distributions Mann-Whitney estimate for the win probability Consider two independent, continuous RVs (random variables) \(\xi\) and \(\eta\). The following probability is called the win probability of RV \(\eta\) against the RV \(\xi\)

Introduction This is a short reminder of some simple properties of exponential distributions. The continuous random variable (RV) \(\xi\) has an exponential distribution with the rate \(\lambda>0\) if its CMD (cumulative distribution function) has the following form