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Finding the pdf of the average of n independent random variables

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data
    The n random variables [itex]X_{1}, X_{2},..., X_{n}[/itex] are mutually independent and distributed with the probability density

    [itex]f(x)=\frac{1}{\pi(1+x^{2})}[/itex]

    i) Find the probability density of the average

    [itex]Y=\frac{1}{n}\Sigma^{i=1}_{n}X_{i}[/itex]

    ii) Explain why it does not converge toward the normal distribution,as would be expected from the central limit theorem.

    2. Relevant equations

    [itex]F_{Y}(y)=\int\int\ldots\int_{Y=\frac{1}{n}\Sigma^{i=1}_{n}X_{i}}f_{X_{1}}(x_{1})f_{X_{2}}(x_{2}) \ldots f_{X_{n}}(x_{n})dx_{1}dx_{2}\ldots dx_{n}[/itex]

    3. The attempt at a solution

    i)
    [itex]F_{Y}(y)=\frac{1}{(2\pi i)^{n}}\int\int\ldots\int_{Y=\frac{1}{n}\Sigma^{i=1}_{n}X_{i}}(\frac{1}{\frac{x_{1}}{n}+i}-\frac{1}{\frac{x_{1}}{n}-i})(\frac{1}{\frac{x_{2}}{n}+i}-\frac{1}{\frac{x_{2}}{n}-i}) \ldots (\frac{1}{\frac{x_{n}}{n}+i}-\frac{1}{\frac{x_{n}}{n}-i})dx_{1}dx_{2}\ldots dx_{n}[/itex]

    That is as far as I got because I don't know how to integrate over a hyperplane...

    ii)
    I Haven't gotten to this but I am guessing that the reason that it does not converge is because the variance is negative or something long those lines.



    I'd appreciate any help, whether there's a simpler way to do part (i) or I did something wrong in it.

    Thanks in advance.
     
  2. jcsd
  3. Feb 1, 2012 #2

    jbunniii

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    Do you know the answer to this general question: If you have two independent random variables, say X and Y, then what is the PDF of their sum X + Y in terms of the PDFs of X and Y?
     
  4. Feb 2, 2012 #3
    do you recognize what this distribution is? and the properties of this distribution?
     
  5. Feb 2, 2012 #4
    jbunniii: Yes I know that it would be the convolution of the pdfs and the integral above can be turned into an convolution integral. What I am saying is that I don't have a clue how to do so for n variables.

    80past2: I honestly have no idea what this distribution is. I haven't taken stat since my freshman year and it really never interested me much, so although I got good grades, my knowledge of statistics is not very thorough. See I was just a math major but I doubled up with physics and am in my 4th year of college right now. Right now I am taking a thermal and statistical mechanics class and this is honestly the first time I have been trumped up on the math in a physics class.
     
  6. Feb 2, 2012 #5
    try it for two and see what happens
     
  7. Feb 2, 2012 #6
    Oh I figured it out it's a cauchy distribution and all I need to do is show that the product of the characteristic equations for x_i/n gives the characteristic equation for y. Thanks for all the help!
     
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