Confusion over using integration to find probability

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Discussion Overview

The discussion centers around the use of integration and cumulative distribution functions (CDFs) in probability, specifically regarding the calculation of probabilities for continuous random variables. Participants explore the application of the complement rule and the interpretation of CDF values in the context of finding probabilities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about why the probability P(X>12) can be calculated as 1-F(12) instead of 1-[F(12) - F(0)], questioning the need to consider the area from 0 to 12.
  • Another participant asks for the value of F(0), indicating a need for clarity on the CDF's behavior at that point.
  • Participants discuss that F(x) represents the probability of a random variable being less than or equal to x, leading to the conclusion that P(X ≤ 12) = F(12).
  • There is a reiteration of the value of F(0) being 0 in this context, which leads to further discussion about the implications for calculating probabilities.
  • One participant reflects on the definition of the CDF and its usefulness, suggesting that F(A) - F(B) represents the area between two points, which is a key aspect of understanding the CDF.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the CDF and the application of the complement rule. There are multiple viewpoints regarding the necessity of considering F(0) in the calculations, and the discussion remains unresolved.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the CDF and the implications of its values at specific points. The relationship between the CDF and the area under the probability density function is also not fully explored.

Of Mike and Men
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Hey everyone, first, let me say I understand the complement rule. Where I am confused is over the integration. My professor said that suppose you have a continuous cumulative distribution function F(x) = 1-e-x/10, if x > 0 (0, otherwise). And suppose you want to find P(X>12) you can use the complement rule 1-P(X<=12). Which is equivalent to 1-F(12) [note he said this works for all cases, not just this example].

My question is why isn't it 1-[F(12) - F(0)]?

This is really tripping me up. If your x can take all probabilities from 0 to 12, don't you want to find the area from 0 to 12 and not just F(12)?

I know this is a method of simplifying the integral since you have an improper integral and have to evaluate a limit (supposing you don't use the compliment rule). But why does this work for all cases?
 
Last edited:
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What value does F(0) take?

Edit: Also, the cdf F(x) is the probability of taking any value less than or equal to x. By definition, you therefore have P(X ≤ 12) = F(12).
 
Orodruin said:
What value does F(0) take?

In this case 1 - 1 = 0. Meaning that you have 1 - [F(12) - 0], in this case. But he said it is true for all cases...
 
Of Mike and Men said:
In this case 1 - 1 = 0. Meaning that you have 1 - F(12) - 0, in this case. But he said it is true for all cases...
Read my edit above.
 
Orodruin said:
Read my edit above.

I guess, why is this the definition? Perhaps this is more calculus related and I'm not remembering. It seems vaguely familiar. I guess it relatively makes sense since F(A) - F(B) would be the area between the two points. Then F(A) would be the entire area up until A since you'd have no lower bound.
 
Of Mike and Men said:
I guess, why is this the definition?
Like most defined things, it is defined because it is useful.

Indeed, F(B)-F(A) would be equal to P(A<X≤B).
 

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