- #1

ChrisVer

Gold Member

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## Main Question or Discussion Point

Hi, I am a little "confused" of how to treat a problem stating: suppose you expect 1.25 background (SM) events from one measurement. During one particular measurement, you observe 5 events. What's the probability that you discovered new physics?

The standard/basic way to go about it is to calculate the significance of your "hypothetized new signal", that is [itex] s = \frac{S}{\sqrt{B}}= \frac{5-1.25}{\sqrt{1.25}}=3.35[/itex], which tells us that the new signal should be 3.35 sigma (standard deviations) away from the expected value to explain the observed events. In probabilities this translates to 99.958% signal exists/ 0.042% it's a statistical fluctuation, right?

My question is that the probabilities in those cases are taken from the standard normal distribution. Wouldn't (for so small expected events) the random variable (observed events) be taken out a Poisson distribution? In explicit wouldn't I have to calculate what's the probability of : [itex]P( x \ge 5 | \lambda =1.25 )\approx 0.9\%[/itex] (being statistical fluctuation) and the complementary for a signal? This would be relatively bad for the significance, as the standard deviation of a Poisson distributed variable is not equal to the square root of the events (for low expected events)...

Thanks.

Maybe this deserves to be moved in the statistics, but please do whatever you find fit.

The standard/basic way to go about it is to calculate the significance of your "hypothetized new signal", that is [itex] s = \frac{S}{\sqrt{B}}= \frac{5-1.25}{\sqrt{1.25}}=3.35[/itex], which tells us that the new signal should be 3.35 sigma (standard deviations) away from the expected value to explain the observed events. In probabilities this translates to 99.958% signal exists/ 0.042% it's a statistical fluctuation, right?

My question is that the probabilities in those cases are taken from the standard normal distribution. Wouldn't (for so small expected events) the random variable (observed events) be taken out a Poisson distribution? In explicit wouldn't I have to calculate what's the probability of : [itex]P( x \ge 5 | \lambda =1.25 )\approx 0.9\%[/itex] (being statistical fluctuation) and the complementary for a signal? This would be relatively bad for the significance, as the standard deviation of a Poisson distributed variable is not equal to the square root of the events (for low expected events)...

Thanks.

Maybe this deserves to be moved in the statistics, but please do whatever you find fit.

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