Confusion over using integration to find probability

In summary, the professor explains that in order to find the probability of X being greater than a certain value, you can use the complement rule of 1 minus the probability of X being less than or equal to that value. This simplifies the integral and works for all cases. The value of F(0) is 0, and the reason for using this definition is because it is useful in finding the area between two points on a continuous cumulative distribution function.
  • #1
Of Mike and Men
54
3
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?
 
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  • #2
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).
 
  • #3
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...
 
  • #4
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.
 
  • #5
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.
 
  • #6
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).
 

FAQ: Confusion over using integration to find probability

What is the purpose of using integration to find probability?

Using integration to find probability is a mathematical technique used to determine the likelihood of a certain event occurring. It involves calculating the area under a probability density function, which represents the probability distribution of a random variable. This method allows for a more accurate and precise calculation of probability compared to other techniques.

How is integration used to find probability?

Integration is used to find probability by calculating the area under a probability density function. This is done by integrating the function over a specific interval, which represents the range of values for the random variable. The resulting value represents the probability of the event occurring within that interval.

Is integration the only way to find probability?

No, integration is not the only way to find probability. There are other techniques such as using the cumulative distribution function or using counting methods, depending on the type of problem. However, integration is often preferred as it allows for more complex and continuous probability distributions to be calculated.

What are some common mistakes when using integration to find probability?

Common mistakes when using integration to find probability include not properly defining the interval of integration, not correctly setting up the probability density function, and not accounting for any constants or transformations in the function. It is important to carefully follow the steps and pay attention to the details when using integration for probability.

Can integration be used for all types of probability distributions?

No, integration cannot be used for all types of probability distributions. Integration is most commonly used for continuous probability distributions, such as the normal distribution, exponential distribution, and uniform distribution. For discrete distributions, other techniques such as counting methods or the cumulative distribution function may be more appropriate.

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