Moments, tails and lack of memory

For the trial-count convention E(X)=1/p and Var(X)=(1-p)/p². The geometric series validates normalisation. For integer k≥0, P(X>k)=(1-p) raised to the power of k ^k. The memoryless property is P(X>s+t|X>s)=P(X>t): after failures, the remaining waiting time has the same distribution. This depends on independence and a constant p.