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

[jimbof85]

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) $\stackrel{95\%}{>}$10)₁₀₀₀

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

Thanks

James

 

[HallsofIvy]

Sounds to me like you are talking about a "95% confidence level".

 

[jimbof85]

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!

 

[Mark44]

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!

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.

 

[jimbof85]

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

 

[jimbof85]

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' : $x\in C$

'natural log tends to infinity as x tends to infinity' : $\lim\limits_{x\to+\infty}$$\ln(x)\to+\infty$

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

Thanks for your help

 

[dodo]

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:$$p(f(x) > 10) = 0.95$$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:$$\hat p_{1000}(f(x) > 10) = 0.95$$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.