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An uncorrected nonnegative symmetric kernel density estimator can leak an order-one amount across a jump in a probability density function. If its transition region has width proportional to the smoothing bandwidth $h$, the integrated mean squared error includes squared bias of an estimator of order $h$, rather than the order $h^4$ familiar for twice-smooth interior densities. For the unit-width uniform window and the exponential distribution of rate one, the integrated squared bias of an estimator is $h/12+o(h)$.

For a nonnegative random variable with finite expected value and $\mathbb{P}(X>t)>0$, its mean residual life is the remaining conditional expected value after surviving past $t$. The tail integral formula for moments gives $m_X(t)={\int }_t^{\infty }\mathbb{P}(X>x)dx/\mathbb{P}(X>t)$. The exponential distribution has constant mean residual life, equal to its original mean. In excess of loss reinsurance, the total variance stationary condition for excess of loss sets a candidate retention equal to this quantity.

An $M/M/s/K$ queue has arrivals forming a Poisson process of rate $\lambda$, $s$ servers with independent service times having an exponential distribution of rate $\mu$, and total system capacity $K\ge s$. Its occupancy is the finite birth-death process on $\{0,\dots ,K\}$ with

For $\log T_z=bz-X$, with $e^X$ unit exponential distribution and $b=\log 2$, the survivor functions are $S_z(t)=1-e^{-2^z/t}$ for $t>0$. These are time-scaled Fréchet distributions, so they form an accelerated life family. However their hazard ratio is $2/(e^{1/t}+1)$, tending to zero as $t\downarrow 0$ and to one as $t\to \infty$. It is not constant. Thus a positive-scale assumption matters when deriving a proportional hazards family from the usual log-location-scale model.

If every holding rate of a continuous-time Markov chain is at most $C<\infty$, its holding times can be coupled as $E_n/q_{Y_n}\ge E_n/C$, where the $E_n$ are independent rate-one exponential random variables. The strong law of large numbers gives ${\sum }_n{E}_n=\infty$ almost surely, so the jump times tend to infinity and the chain cannot explode.