Consider a branching process with a geometric offspring distribution \( P(X=k) = (1-p)p^k\), for \(k=0,1,2,\dots\) . Show that the ultimate extinction is certain if ...

Let \(X_1\) and \(X_2\) be random variables and let \(M=\mathrm{max}(X_1,X_2)\) and \(N=\mathrm{min}(X_1,X_2)\). Argue that the event \(\{ M \leq x\}\) is the same as ...

Researchers have proposed an optimized ratio-type estimator that uses auxiliary attributes to estimate population means more ...