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Maths statement for point when condition is met some fraction of the time

  1. Jan 29, 2013 #1
    Hello all,

    I would like to express the following as an equation, but don't know the nomenclature.

    'The point at which a condition is true 95% of the time'

    ie. I have a function, f(x) which returns some value in the presence of random and uncharicterizable noise. I run this 1000 times. I find the condition f(x)>10 is true 50% of the time. I adjust f(x), and rerun 1000 times and find f(x)>10 is true 80%. I keep rejecting f(x) until I reach the point where f(x)>10 for 95% of samples.

    Basically I want something like (f(x) [itex]\stackrel{95\%}{>}[/itex]10)1000

    but there is bound to be a correct way to do this

    Thanks

    James
     
  2. jcsd
  3. Jan 29, 2013 #2

    HallsofIvy

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    Science Advisor

    Sounds to me like you are talking about a "95% confidence level".
     
  4. Jan 29, 2013 #3
    Hi HallsofIvy,

    Thanks for the reply. It is not (I believe) fair to talk about confidence levels in the way I think you suggest - this is not a normally distributed random variable, it is not a two tailed distribution. I am in fact fitting a model to my data to get a chi sqrd, comparing this chi sqrd to a threshold, and if it is below this claim a success. I simply extract the point where some percentage of these results are successes - I chose 95%. The distribution of the number of successes in each of the many variations of the experiment is not normally distributed. I simply want an mathematical expression that says the equivalent of 'x > y 95% of the time' rather than 'x>y'

    Sorry i can't put it clearer than that!
     
  5. Jan 29, 2013 #4

    Mark44

    Staff: Mentor

    Confidence intervals are not limited just to normal distributions. This concept can be applied to ##\chi^2## distributions for inferences about ##\sigma^2 ##. The hypothesis tests can be one-tailed or two-tailed.
     
  6. Jan 29, 2013 #5
    Hi Mark44,

    Thanks for the reply. Ok, I understand your point, but i am not asking for a confidence interval. My question is on nomenclature. I simply need a mathematical way of conveying the following statement 'f(x) > y 95% of the time'. I do not need the distribution behind it, or the mathematics that control it - I am just after the correct symbols to properly convey that I have chosen a set of values for f(x) such that it meets some criterion for some fraction of realisations of the experiment.

    Thanks
     
  7. Jan 29, 2013 #6
    Just to further clarify, I simply want to do as follows, take an expression in words and write it using mathematical notation. i.e.

    'x is a complex number' : [itex]x\in C[/itex]

    'natural log tends to infinity as x tends to infinity' : [itex]\lim\limits_{x\to+\infty}[/itex][itex]\ln(x)\to+\infty[/itex]

    'f(x) > y 95% of the time' : ???

    Thanks for your help
     
  8. Jan 29, 2013 #7
    It seems to me that, if P(E) denotes the probability of an event E, you are referring to an estimation of the following statement:[tex]p(f(x) > 10) = 0.95[/tex]This statement, however, speaks of a theoretical (and unknown) probability, which your experiment is trying to estimate. An often used notation is to write a caret or "hat" symbol over the letter, to denote it is an estimation:[tex]\hat p_{1000}(f(x) > 10) = 0.95[/tex]But I think it is unavoidable to accompany these lines with a few words in plain English (as I did) that define what p-hat means in your context. My 2 cents.
     
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