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Heaviside Function

  1. Sep 20, 2011 #1
    Hi all:

    The Heaviside function multiples a random varaible, is that a probability density function?

    This is my first time knowing about Heaviside, any tutorial and application of it?
  2. jcsd
  3. Sep 20, 2011 #2
    The Heaviside function multiplied by a random variable (that may only have a real value) is not a probability density function. For a probability density function P of a random variable which may only have real values,

    [itex]\int _{\infty}^{\infty} P(x)dx = 1[/itex]

    whereas if H(x) is the Heaviside step function, and a is a real number

    [itex]\int _{\infty}^{\infty} aH(x)dx \neq 1[/itex]
  4. Sep 21, 2011 #3


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    Science Advisor

    You need to normalize your pdf.

    This kind of thing happens in Bayesian statistics. What happens is you have a prior and a likelihood and then you create the posterior from the prior and the likelihood.

    One thing you should realize though is that if you want to use the Heaviside function as some kind of prior, you will need to normalize the posterior. What will happen is that for a probability to be normalized, if you have a likelihood that is normalized and a prior that is not (this will definitely nearly always be the case), then you need to find the total area of the un-normalized pdf and divide your posterior definition by this area.

    In terms of interpreting what you are doing, it is basically the equivalent of defining a uniform prior in some (possibly collections of) interval(s).

    So yeah, figuring out the total area of our new pdf (find the integral over the whole real line for your new pdf), and divide your pdf by that number and you will have a proper pdf.
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