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Math Help - I need help making up a skewed PDF

  1. #1
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    I need help making up a skewed PDF

    Basically I need to make up a PDF that is skewed enough that it wont really fit the CLT unless I have a large number of observations. I have no idea how to just make one up (I cant use anything with a name like Chi squared, poisson, etc.)

    I know obviously the properties of a PDF (it must have all positive values, the integral has to equal 1, etc.)

    I just need some help on how to start.

    Thanks in advance
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  2. #2
    Senior Member Sambit's Avatar
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    You may construct a discrete probability distribution with more frequencies on the left than on the right.
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  3. #3
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    the problem is construct out of what? I have never had to do anything but use a given PDF to find this probability etc. I dont know how to create a PDF from scratch.
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  4. #4
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    Consider densities of the form:

    f(x) = \theta x ^{-\theta - 1}I(x \ge 1)

    (Note: I came up with this on my own, but after a few minutes I realized this is a Pareto distribution. So, it has a name, but it is very easy to come up with on your own so I would think it still counts since the Pareto isn't usually covered extensively).

    As far as coming up with your own pdfs: every function that is positive and has a finite integral - i.e. \int_{-\infty}^ \infty f(x) \ dx < \infty - can be turned into a pdf by dividing by the value of the integral. So, pick a function that integrates and then normalize it to get a pdf.
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  5. #5
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    okay first off what is I? Second, when you say I came up with this on my own, the question is how? What made you think of that?
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  6. #6
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    I(\cdot) is the indicator function. It is equal to 1 if the statement inside is true, and 0 otherwise. It is a more compact way of writing

    f(x) = \begin{cases} \theta x ^ {-\theta - 1} \qquad & x \ge 1 \\ 0 \qquad & x < 1 \end{cases}

    I came up with it by thinking of a function that was integrable on the whole real line, but did not satisfy \mbox{E} X ^ p < \infty for all p. So, for appropriate choice of theta, the mean, variance, skewness, kurtosis, etc. may or may not exist. This gives me a good deal of control over how long it takes the central limit theorem to work. Of course, it probably wouldn't be reasonable to expect a student to think of all those things at once, but it IS skewed for appropriately chosen theta which is something you could notice (I won't tell you which ones, though, because I should at least leave you some work to do).

    A word of caution if you decide to use this: the choice of theta is important. If you choose theta too small the CLT won't apply at all - obviously theta needs to be positive, but that isn't the end of it if you want to claim the CLT holds.
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  7. #7
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    Um so in terms of coming up with my own, when you say divide it by the value of its integral, so you take the function Y, then integrate it from -infinity to inifinity and that is the pdf?
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  8. #8
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    for example how would you make e^x a pdf?

    or could you not since the integral is e^x?
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  9. #9
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    Quote Originally Posted by xamius View Post
    for example how would you make e^x a pdf?

    or could you not since the integral is e^x?
    To turn f(x) into a pdf, just take

    \displaystyle<br />
p(x) = \frac{f(x)}{\int_{-\infty} ^ \infty f(x) \ dx}.<br />

    This doesn't work for e^x because \int_{-\infty}^\infty e^x \ dx = \infty.
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  10. #10
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    so for example the pdf of the function 2x^2 is 3/x?
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  11. #11
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    It's a definite integral...
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