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Distribution of weighted Normal Distributions

  1. Sep 15, 2011 #1
    You created a random number generator that works as follows:
    With probability p it selects a number X from the standard normal distribution N(0,1), and
    with complimentary probability (1-p) it selects a random number X from an off-central
    normal distribution N(5, 1). Write the distribution function of X.

    How would you attempt this.
    Obviously the variance increases.
    and the mean is a weighted average of the two.
    but as far as getting fx(x) I am stumped.
    Is it correct to add the two distributions together and simplify?
    (p)*N(0,1) + (1-p)*N(5,1)
    using the gaussian equation?
  2. jcsd
  3. Sep 15, 2011 #2


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    looks reasonable to me - when you have a discrete probability to realise a given distribution, you can just add the distributions together weighted by their probabilties

    if it doubt try it, this would be pretty simple to implement in excel as a check
  4. Sep 15, 2011 #3

    Ray Vickson

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    Yes, it is correct. However, it won't simplify. The final result is NOT itself a Gaussian, or anything like it.

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