Renewal interval containing a fixed time

The renewal interval containing $t$ has lengthSelection by a fixed observation time favours longer interarrival times.

For a nonnegative random variable $X$ with $0<\mathbb{E}X<\infty$, its size-biased version $\hat{X}$ is defined byfor bounded measurable $h$. If $X$ has density $f$, then $\hat{X}$ has density $xf(x)/\mathbb{E}X$.

Reweight probabilities proportionally to the nonnegative value being observed, then divide by its positive finite expected value. This produces a size-biased distribution. Values twice as large receive twice the original relative weight; this is a change in the sampling law rather than a deterministic multiplication of the sampled value.

For the positive-support geometric distribution with success probability $p$, size bias gives $\mathbb{P}(\tilde{Y}=k)=k{p}^2(1-p{)}^{k-1}$. The sum of two independent copies has $k$ positive-integer decompositions at value $k+1$, so its shifted probability is exactly this expression. Its probability generating function is $p^{2}s/(1-(1-p)s{)}^2$, also obtained by the size-biased probability generating function identity.

For a positive integer-valued random variable with finite mean $\mu$, the size-biased distribution has probabilities $k\mathbb{P}(Y=k)/\mu$. They sum to one by the definition of expected value. Multiplying the differentiated probability generating function by $s/\mu$ gives the displayed identity for $0\le s<1$, with the endpoint interpreted by its limit.

For exponential interarrival times of rate $\lambda$, the age and residual life at time $t$ converge jointly to independent rate-$\lambda$ exponential variables. Their sum therefore converges to the gamma distribution with density ${\lambda }^{2}x{e}^{-\lambda x}$, the size-biased exponential law.

At every deterministic $t>0$, the containing interval stochastically dominates an unselected interarrival time:Conditioning on the first interarrival gives a renewal equation; subtracting the constant tail probability leaves a renewal equation with a nonnegative forcing term.