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Math Help - 2nd continuous random variable

  1. #1
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    2nd continuous random variable

    Consider the continuous random variable X with probability density function as:


    Find
    a) the value of C
    b) the distribution function of X
    c) the second moment about the origin of X.

    Again, infinity confuses me. I can't figure out how to do the problem because it's not a "normal" number.
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  2. #2
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    Quote Originally Posted by ban26ana View Post
    Consider the continuous random variable X with probability density function as:


    Find
    a) the value of C
    b) the distribution function of X
    c) the second moment about the origin of X.

    Again, infinity confuses me. I can't figure out how to do the problem because it's not a "normal" number.
    a) \int_{-\infty}^{+\infty} \frac{C}{1 + x^2} \, dx = \lim_{\alpha \rightarrow \infty} \left[ C \tan^{-1} x\right]_{-\alpha}^{\alpha} = C \pi \, ....

    b) Do you mean the cumulative density function F(x) = \Pr(X < x).

    c) Try to calculate E(X^2). You'll find it's undefined. The mean (first moment about the origin) of X is also undefined. This is a famous property of this particular pdf (which is an example of a Cauchy distribution: http://en.wikipedia.org/wiki/Cauchy_distribution).
    Last edited by mr fantastic; October 2nd 2008 at 04:09 AM. Reason: Fixed a typesetting error
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  3. #3
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    [quote=mr fantastic;196408]
    b) Do you mean the cumulative density function F(x) = \Pr(X < x)>
    [quote]
    I assume so. Our professor calls it the distribution function.
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  4. #4
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    Quote Originally Posted by ban26ana View Post
    Quote Originally Posted by mr fantastic View Post
    b) Do you mean the cumulative density function F(x) = \Pr(X < x)
    I assume so. Our professor calls it the distribution function.
    F(x) = \Pr(X < x) = \frac{1}{\pi} \int_{-\infty}^{x} \frac{1}{1 + t^2} \, dt = \frac{1}{\pi} \lim_{\alpha \rightarrow - \infty} \left[ \tan^{-1} t \right]_{\alpha}^{x} = \, ....
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