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A Deriving the standard normal distribution

  1. Apr 9, 2016 #1
    I've calculated the joint distribution, XY_PDF(x,y) of random
    variables X and Y (both coming from a distribution N(n) = C*e^(-K*n^2)).

    I use XY_PDF(x,y) to calculate the joint distribution AR_PDF(a,r)
    of the random variables A (angle) and R (radius), with the PDF
    method and the Jacobian.

    Since AR_PDF(a,r) = A_PDF(a)*R_PDF(r) and I want A_PDF(a) = 1/(2*pi),
    I can find that C = 1/(2*pi)^(1/2) in N(n), since the other factors in
    AR_PDF(a,r) calculated from XY_PDF(x,y) are related to R.

    If I do the same in 3D (using the longitude/latitude/radius distributions
    for producing a uniform surface distribution), I get C = 1/(2*pi)^(1/3)
    after discarding the factors related to the longitude distribution and the
    radius distribution.

    The correct answer (for a multivariate gaussian) should be 1/(2*pi)^(1/2)
    here also, right?

    Is my reasoning to find this constant C wrong? Is there a better way?
     
    Last edited: Apr 9, 2016
  2. jcsd
  3. Apr 9, 2016 #2

    andrewkirk

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    You don't need to perform the Cartesian to Polar transformation to calculate C. Just observe that the pdf N is that of a normal random variable with zero mean and variance ##\sigma^2=\frac{1}{2K}##. Then use the formula for a normal distribution to conclude that
    $$C=\frac{1}{\sigma\sqrt{2\pi}}=\sqrt\frac{2K}{2\pi}$$
    The formula you calculated above is missing the factor ##\sqrt{2K}##.
     
  4. Apr 10, 2016 #3
    Is there no way to derive the standard normal distribution (not the normal distribution with concepts like mean and variance) from just basic knowledge of:
    - The Jacobian
    - How to calculate one joint distribution from another joint distribution using the PDF method
    - Random vectors in 2D/3D (deriving the distributions of independent variables (polar/spherical coordinates) to get a uniform surface distribution by calculating the joint distribution of their dependent cartesian coordinates)
    - Isotropy, wanting another kind of random vector by letting the cartesian coordinates be independent by coming from the same unknown distribution, N, as the radius
     
  5. Apr 12, 2016 #4
    For a standard normal distribution in n dimensional Euclidean space Rn, I like to think of it just as the joint distribution of n independent 1-dimensional standard normal random variables. As such, its n-dimensional probability density function is just the product

    (1/√(2π)) e-x12/2 ⋅ ... ⋅ (1/√(2π)) e-xn2/2

    which is

    (2π)-n/2 e-(x12 + ... + xn2)/2

    or in other words

    (2π)-n/2 e-r2/2,

    where r is the distance from the origin in Euclidean n-space Rn.

    I know this doesn't answer all parts of your question, but it's one easy way to remember the constant factor occurring in the n-dimensional standard normal density.
     
  6. Apr 12, 2016 #5
    Thanks, zinq
     
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