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Thread: Question on pdf and cdf for unspecified density function

  1. #1
    Senior Member dokrbb's Avatar
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    Question on pdf and cdf for unspecified density function

    I'll bug you guys until I get comfortable enough with all these techniques.

    So, I have a cdf $F_{K}$ for a random variable K, which is strictly increasing and continuous function over the entire space of reals.

    In order to find the distribution of $F(K)$ I define another random variable as a one-to-one correspondence $M = g(K)$, where $g^{-1}$ is also continuous, increasing over all the real space.

    Then, by def of cdf, I have:
    $F_{M}(m) = P[M \leq m] = P[g(K) \leq m] = P[K \leq g^{-1}(m)] = F_{K}(g^{-1}(m)).$

    Thus, the distribution of the $F_{K}$ is defined as follows:

    if $g^{-1}(m) \leq 0 , F_{K} = 0$
    if $0 < g^{-1}(m) < 1 , F_{K} = k$
    if $1 \leq g^{-1}(m) , F_{K} = 1$.

    Kind of makes sense, although if there is a simpler way to define the distribution let me know.

    Now, the actual question comes after - assuming that the RV K has a density function $f$, then I need to find the pmf of the $\lceil K \rceil$ (in the sense that for $k = 0.1$, $\lceil k \rceil = 1$, as is the case for the ceiling function).

    I know that a pmf of a discrete random variable Y is defined as $P(Y = y)$, but how to transform this continues (or real-valued) K into a discrete and make up a pmf for it, confuses me a bit.

    Could you please help me here

    Thanks a lot
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    Re: Question on pdf and cdf for unspecified density function

    you've thrown a bunch of stuff together here

    let's say you have a random variable $K$ that has a probability density function $f_K(x)$

    You want to find the PMF of $Y=\left \lceil K \right \rceil$

    The way I'd look at this is that

    $Y=k \Rightarrow k-1 < K \leq k,~k\in \mathbb{Z}$

    thus

    $P[Y=k] = F_K(k)-F_K(k-1)$

    and that's your PMF for $Y$
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    Re: Question on pdf and cdf for unspecified density function

    Hey dokrbb.

    Hint - What do you think for your g() function will be in this case? Also - does it satisfy the conditions laid out in your original post to be used for that result?
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  4. #4
    Senior Member dokrbb's Avatar
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    Re: Question on pdf and cdf for unspecified density function

    romsek, I'm supposed to show my work, right?

    And shouldn't my answer be defined in integral terms, for completeness?

    $ P[Y = k] = F(k) - F(k-1) = \int_{k}^{k-1}f(u)du = - \int_{k-1}^{k}f(u)du $
    Last edited by dokrbb; Mar 14th 2017 at 04:06 AM.
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    Senior Member dokrbb's Avatar
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    Re: Question on pdf and cdf for unspecified density function

    chiro, I suppose my $g$ function should be a bijective function, that is, one-to-one and onto. What else do I miss in defining the $g()$ function.
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    Re: Question on pdf and cdf for unspecified density function

    Quote Originally Posted by dokrbb View Post
    romsek, I'm supposed to show my work, right?

    And shouldn't my answer be defined in integral terms, for completeness?

    $ P[Y = k] = F(k) - F(k-1) = \int_{k}^{k-1}f(u)du = - \int_{k-1}^{k}f(u)du $
    eh not really. You're given some $F_K$ in this problem.

    In the last problem we needed to use the integrals because the property we were given was in terms of $f_K$

    Either answer should be fine.
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    Re: Question on pdf and cdf for unspecified density function

    Does the ceiling function have the properties listed in your original post?
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  8. #8
    Senior Member dokrbb's Avatar
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    Re: Question on pdf and cdf for unspecified density function

    hey chiro,

    No, definitely not - the ceiling function is discrete (not continuous) and not one-to-one (it's mapping many real RVs between (k-1) and k to (k-1));

    So, can you tell me where you want to get? Do you mean that for the first part I should be more specific and define $g$ somehow specifically, or on a specific interval?
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    Re: Question on pdf and cdf for unspecified density function

    That means that the result won't be able to be used.

    If you ask these kinds of questions yourself it should help you in your work.

    I'm not trying to make you have a right or wrong answer - but to think about what you're doing and see what conclusions follow.

    If you can't find a way to use the result then you will have to get some other method. If the question asks you to use the result then the answer should be that you can't.
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  10. #10
    Senior Member dokrbb's Avatar
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    Re: Question on pdf and cdf for unspecified density function

    oh, in this sense - well, since on the interval [0, 1], my inverse of g function gives the value of y, I will have a Uniform distribution over this this interval. Now I can use the result. Although, for these kind of questions, I am not always sure whether to try to get the most out of it or keep the whole answer as a symbolic expression, since the question is mostly symbolic.
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