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Math Help - help me to convolute non central chi-square with white Gaussian distribution

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    help me to convolute non central chi-square with white Gaussian distribution

    Hi, I am doing research and i am stuck at this point I need help to convolute non central chi-square with white Gaussian distribution.
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    Senior Member MaxJasper's Avatar
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    Re: help me to convolute non central chi-square with white Gaussian distribution

    Do you need approximation or exact formulas?
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    Re: help me to convolute non central chi-square with white Gaussian distribution

    Thanks for your reply I really appreciate your help.......approximation will be OK if you don't know the exact formula....
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: help me to convolute non central chi-square with white Gaussian distribution

    Non-Central \chi ^2 Distribution in terms of \chi ^2 Distribution:

    P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\text{:=}\sum _{j=0}^{\infty } \frac{e^{-\frac{\lambda }{2}} \left(\frac{\lambda }{2}\right)^j P\left(\left.\left(\chi '\right)^2\right|2 j+\nu \right)}{j!}

    \lambda \geq 0 non-centrality parameter.

    Non-Central \chi ^2 Distribution in terms of Normal Distribution:

    \to means approximately equal.

    P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\to P(x)

    x\text{:=}\frac{\sqrt[3]{\frac{\left(\chi '\right)^2}{a}}-\left(1-\frac{2 (b+1)}{9 a}\right)}{\sqrt{\frac{2 (b+1)}{9 a}}}

    A simpler approximation:

    P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\to P(x)

    x\text{:=}\sqrt{\frac{2 \left(\chi '\right)^2}{b+1}}-\sqrt{\frac{2 a}{b+1}-1}
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    Re: help me to convolute non central chi-square with white Gaussian distribution

    Thank you for your help.....can I know what is end result of the convolution.
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: help me to convolute non central chi-square with white Gaussian distribution

    In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form

    \sum_{i=1}^n X_i \sim Y

    where X_1, X_2,\dots, X_n\, are independent and identically distributed random variables. In place of X_i and Y the names of the corresponding distributions and their parameters have been indicated.

    List of convolutions of probability distributions - Wikipedia, the free encyclopedia
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