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If the cumulative distribution function of a random variable is

$$F(x) = P(X\leq x)$$

how can this be transformed mathematically to, and the meaning of

$$1-F(x)$$

develarist
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  • Your question is not clear – gunes Aug 30 '20 at 17:03
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    The meaning of $1-F(x)$ is $P(X>x)$: $P(X\le x \text{ or }X>x)=F(x)+(1-F(x))=1$. – Sergio Aug 30 '20 at 17:04
  • @Sergio thanks for the derivation. is the meaning that $1-F(X)$ is just the other 'half' of the CDF? and what is the condition after $:$ saying? it just ensures that the CDF and its 'other half' sum to 1? – develarist Aug 30 '20 at 17:08
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    @develarist Yes: $P(X>x)=1-P(X\le x)=1-F(x)$. – Sergio Aug 30 '20 at 17:10
  • Here is a related question: https://stats.stackexchange.com/questions/485332/derivation-of-survival-copula – develarist Aug 30 '20 at 17:31
  • $R_X(t) = P(X > t) = 1 - P(X \le t) = 1 -F_X(t)$ is sometimes called the _reliability_ function of random variable $X.$ – BruceET Aug 31 '20 at 01:21

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The transformation $1-F(x)$ of a distribution function $F(x)$ is often called the survival function or reliability function of the distribution. This has the most obvious interpretation when $x$ represents time and $F(x)$ is the cumulative probability of an event occurring before or at time $x$. Then the survival function is the probability of having survived longer than $x$. There is, however, no need to restrict usage of that term to survival or reliability analysis.

EdM
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  • what other applications are out there where time is not used for $x$? – develarist Aug 30 '20 at 17:32
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    @develarist in principle, you could use $x$ to represent extensive properties like distance, intensive properties like temperature, count values like numbers of web site visits, etc. The survival function is useful when your primary interest is in not yet having experienced the event when some distance, temperature, number of web site visits, has been reached. – EdM Aug 30 '20 at 17:42
  • in the context of not yet having experienced a rare event, these other models then use $1-F(x)$ as some sort of anticipation function then? how else can we label this anticipation concept – develarist Aug 30 '20 at 17:48
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    @develarist the "survival function" terminology is still useful in those contexts, it's just that there's been "survival" beyond a certain extent of space, level of temperature, number of web visits, etc. Technical use of that terminology should be readily understood in those other contexts. Speaking as someone pretty far along a personal survival curve, "anticipation" is not the terminology that I would choose to use about the event that has not yet occurred. – EdM Aug 30 '20 at 17:54
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    Other names I've seen are reverse, converse or complementary distribution functions, reliability function and survivor function. Note as an important nuance that working instead with $P(X \ge x)$ is a popular convention; then all data values can be shown on a graph showing probabilities on a logarithmic scale. (Alternatively in practice people may work with plotting positions such as (rank $-\ a$) / (sample size $-\ 2a + 1$) e.g. $a = 0.5$.) – Nick Cox Aug 30 '20 at 18:00
  • What is the probability of a yet more extreme event is a central (even the central) question in looking at disasters of various kinds (floods, droughts, earthquakes, eruptions, financial crashes, etc.) which makes this way of thinking about distributions natural as well as convenient. – Nick Cox Aug 30 '20 at 18:03