Probability of two events in time series: Which event is more likely to occur?

Question

Let $P_i(t) \in [0, 1]$ be probability of event $i = 1, 2$ occur at time $t \geq 0$, both of which satisfy $P_i(\infty) = 0$, and $\int_0^\infty P_i(t) \, dt < \infty$.

I would like to know which event is more likely to occur over the infinite time horizon. Does the following ratio make sense for evaluation? $$ \frac{\mathbb{E}[P_1(t)]}{\mathbb{E}[P_2(t)]} = \frac{\int_0^\infty P_1(t) \, dt}{\int_0^\infty P_2(t) \, dt} $$

Does the ratio $\mathbb{E}[P_1(t)] \, / \, \mathbb{E}[P_2(t)]$ make sense because $P_i(t)$ are not random variables?

Or directly, is $\bigl(\int_0^\infty P_1(t) \, dt\bigr) \, / \, \bigl(\int_0^\infty P_2(t) \, dt\bigr)$ appropriate?

Answer 1

• If $\ T_i\ $ is the time when event $i$ occurs, then $\ T_i\ $ is a random variable with $\ P_i(t)=\lim_\limits{x\rightarrow t-}P\left(x<T_i\le t\right)\ $, which cannot be non-zero at more than a countable number of points. Therefore $\ \int_\limits{0}^\infty P_i(t)\,dt=0\ $, and the expression $$ \frac{\int_\limits{0}^\infty P_1(t)\,dt}{\int_\limits{0}^\infty P_2(t)\,dt} $$ is undefined.
• As long as $\ P_2(t)\ne 0\ $ the ratio $\ \frac{\mathbb{E}\left(P_1(t)\right)}{\mathbb{E}\left(P_2(t)\right)}\ $ is a well-defined quantity. However, since (as you have realised) neither $\ P_1(t)\ $ nor $\ P_2(t)\ $ are random variables, $\ \mathbb{E}\left(P_i(t)\right)\ $ is simply equal to $\ P_i(t)\ $ (not $\ \int_\limits{0}^\infty P_i(t)\,dt\ $) and therefore $$ \frac{\mathbb{E}\left(P_1(t)\right)}{\mathbb{E}\left(P_2(t)\right)}= \frac{P_1(t)}{P_2(t)}\ . $$ Note, however that $\ P_2(t)\ $ can only be non-zero at no more than a countable number of values of $\ t\ $, and will be identically zero if $\ T_2\ $ has a continuous distribution.
• The only reasonable meaning I can think of for the expression $\ P_i(\infty)\ $ is $\ \lim_\limits{t\rightarrow\infty}P\left(T_i>t\right)\ $, which is the probability that event $\ i\ $ never occurs. If this is not what you intended it to mean, you will need to explain exactly what you did intend it to mean. But if $\ P_i(\infty)\ =\lim_\limits{t\rightarrow\infty}P\left(T_i>t\right)\ $ is the correct interpretation, then $\ P_1(\infty)= P_2(\infty)=0\ $ means that both events $1$ and $2$ are certain to occur at some finite time, so neither is more likely than the other to occur "over the infinite time horizon".