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Homework Help: 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


    i) Find the probability density of the average


    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

    [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...

    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


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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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