论文标题
超级悖论
Supercentenarian paradox
论文作者
论文摘要
考虑以下声明： $ b（t，Δt）$：a $ t $年龄的人NN将在$ΔT$年中幸存下来，其中$ t，Δt\ in \ mathbb {r} $是非负实数。我们只知道NN已有$ t $年的历史，而与健康状况，性别，种族，国籍等无关。我们敢打赌，$ b（t，Δt）$所持。看来我们的赔率非常好，对于提供$ΔT$的任何$ t $来说，$ 1/365美元（即一天）。但是，这并不是那么明显，并且取决于寿命概率分布。令$ f（t）$表示最多$ t $年的生活的可能性，并设置$φ（t）= 1 -f（t）$。 显然，$φ（t）\ rightarrow 0 $ as $ t \ rightarrow \ infty $。对于任何固定的$ΔT$，$ pr（b（t，Δt））\ rightArrow 0 $ as对于任何固定的$ΔT$，每当$δT$（$φφ$的收敛都足够快）（例如，超级指数）时，不难验证$ pr（b（t，Δt））\ rightArrow 0 $。统计数据提供了这种情况的参数（基于推断）。因此，对于任意小的正$ΔT$和$ε$，存在足够大的$ t $，因此$ pr（b（t，Δt））<ε$，这意味着我们不应该在理论上……。但是，实际上，我们可以安全地押注，因为对于不平等$ pr（b（t，Δt））<1/2 $ <1/2 $ a需要非常大的$ t $。例如，$ΔT= 1/365美元可能需要$ t> 125 $ $年，对于文献中考虑的某些典型分布$ f $。然而，在地球上没有这样的人。因此，我们的赔率是好的，要么是因为所选的测试者nn还不够大，要么出于技术（或更确切地说，是统计）原因 - 缺乏测试者。这种情况类似于著名的圣佩特斯堡悖论。
Consider the following statement: $B(t, Δt)$: a $t$ years old person NN will survive another $Δt$ years, where $t, Δt\in \mathbb{R}$ are nonnegative real numbers. We know only that NN is $t$ years old and nothing about the health conditions, gender, race, nationality, etc. We bet that $B(t, Δt)$ holds. It seems that our odds are very good, for any $t$ provided $Δt$ is small enough, say, $1 / 365$ (that is, one day). However, this is not that obvious and depends on the life-time probabilistic distribution. Let $F(t)$ denote the probability to live at most $t$ years and set $Φ(t) = 1 - F(t)$. Clearly, $Φ(t) \rightarrow 0$ as $t \rightarrow \infty$. It is not difficult to verify that $Pr(B(t, Δt)) \rightarrow 0$ as $t \rightarrow \infty$, for any fixed $Δt$, whenever the convergence of $Φ$ is fast enough (say, super-exponential). Statistics provide arguments (based on an extrapolation yet) that this is the case. Hence, for an arbitrarily small positive $Δt $ and $ε$ there exists a sufficiently large $t$ such that $Pr(B(t, Δt)) < ε$, which means that we should not bet... in theory. However, in practice we can bet safely, because for the inequality $Pr(B(t, Δt)) < 1/2$ a very large $t$ is required. For example, $Δt = 1/365$ may require $t > 125$ years for some typical distributions $F$ considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good, either because the chosen testee NN is not old enough, or for technical (or, more precisely, statistical) reasons -- absence of a testee. This situation is similar to the famous St.Petersburg Paradox.
