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Sum of noncentral chi-square RVs

  1. Jan 30, 2010 #1
    Hi guys,

    i amtrying to find the sum of N random variables each follow the noncentral chi-square distribution and they are i.i.d, i.e,

    sum(y_i), i=1,...N

    y_i is the RV and has a noncentral chi-square pdf

    f[y](y) = (exp(-(H[i, d]+y)/sigma^2)*BesselJ(0, 2*sqrt(H[i, d]*y/sigma^4))/sigma^2

    plz help me
  2. jcsd
  3. Jan 31, 2010 #2


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    A sum of independent noncentral chi-square RVs is noncentral chi-square. You just add their degrees of freedom and means together. This follows directly from the definition (depending what definition you are using for chi-square, of course).
  4. Jan 31, 2010 #3
    Hi, thanks for the info.
    Actually, i cannot understand what do you mean by chi-square definition, my noncentral chi-square is a result of squared Gaussian RV with mean and segma^2.

    i found an expression of the characteristic function of the noncentral chi-square but i am not sure if this expression is sufficient to represent the sums of the noncentral chi-square distribution, i am also not sure how to simulate this formula in matlab for example, can you help in that, or do you have any comment

  5. Feb 2, 2010 #4


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    Homework Helper

    The characteristic function approach will work: If [tex] X_1, X_2, \dots, X_n [/tex]
    are any independent rvs, then the characteristic function [tex] X_1 + X_2 + \dots + X_n [/tex] equals the product of the individual characteristic functions: write an expression for the product of the c.v.s in your problem, note its form and what it tells you about the distribution of the corresponding sum.

    I'm not sure what you mean by 'simulate in matlab'.
  6. Feb 2, 2010 #5


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    A sum of m independent unit variance squared Gaussian RVs added to a sum of n independent unit variance squared Gaussian RVs is clearly equal to a sum of (m+n) such squared Gaussians. So the sum of independent chi square RVs with degrees of freedom m and n respectively is a chi square with (m+n) degrees of freedom.
    However, you seem to have an additional scaling factor sigma, in which case this will not hold if they have different sigmas.
  7. Feb 2, 2010 #6
    thanks guys,
    I found the analysis in one book, The Algebra of Random variables, M. D. Springer, university of Arkansas, it is a very old book but really good one one. now i am trying to find the ratio of two i.r.v each follow noncentral chi-square distribution
    Any idea about that,
  8. Feb 3, 2010 #7
    Look up the F distribution. It may not be in older textbooks, but there are good descriptions on the web.
  9. Jan 30, 2012 #8
    Hi Guys,

    By definition, the sum of iid non-central chi-square RVs is non-central chi-square. what is the sum of ono-identical non-central chi-square RV.

    I have a set of non zero mean complex Gaussian random variables H_i with a mean m_i and variance σ_i . i=1...N. H
    the result of their square is non-central chi-square RM. Now what is the distribution of the sum of those non-central chi-square RV given that their variances are different "i.e., they are independent but non-identical distributed".

    Kind Regards
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