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B Probability density of a normal distribution

  1. Jan 6, 2017 #1
    If the normalized probability density of the normal distribution is ## p(x) = \frac {1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} ##, then if ##\sigma = 0.0001## and in the special case ## x = \mu##, wouldn't the probability density at this point, ##p(\mu)##, exceed 1 since it is equal to ##p(\mu) = \frac {1}{\sqrt{2\pi}0.0001} > 1##? Wouldn't this mean it is not normalized?
     
  2. jcsd
  3. Jan 6, 2017 #2

    FactChecker

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    No. The density function can get huge as long as its integral is equal to 1. So the density function can get very large for a short range of X.
     
  4. Jan 6, 2017 #3
    Okay, just wanted to ensure I understood that. Thank you. So in general, the value for ##p(x)## always varies from 0 to ##\frac{1}{\sqrt{2\pi}\sigma}##?
     
  5. Jan 6, 2017 #4

    mfb

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    Yes.

    Other probability distributions can have even higher densities - as long as they are in a small range. Only the integral is important.
     
  6. Jan 7, 2017 #5

    Stephen Tashi

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    As an analogy, a 1 kg rock with a volume of 300 cc could have a density of 1.5 kg/ 300 cc at a particular location within the rock.
     
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