Find the first moment of a probability distribution governed by a nonlinear first order ODE

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May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example,
$$
fracmathrm d p(x)mathrm dx = alpha(x) p(x)
$$
where $alpha(x)$ is a non-linear function in terms of $x$. As far as I know, the most straightforward method is trying to solve the distribution $p(x)$ directly, and then find the first moment by
$$
langle xrangle = int x p(x)mathrm dx
$$
but sometimes the analytical form of this distribution is very hard to obtain, so instead of doing this is there other method?




probability differential-equations moment-generating-functions




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    up vote
    2
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    May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example,
    $$
    fracmathrm d p(x)mathrm dx = alpha(x) p(x)
    $$
    where $alpha(x)$ is a non-linear function in terms of $x$. As far as I know, the most straightforward method is trying to solve the distribution $p(x)$ directly, and then find the first moment by
    $$
    langle xrangle = int x p(x)mathrm dx
    $$
    but sometimes the analytical form of this distribution is very hard to obtain, so instead of doing this is there other method?




    probability differential-equations moment-generating-functions




    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example,
      $$
      fracmathrm d p(x)mathrm dx = alpha(x) p(x)
      $$
      where $alpha(x)$ is a non-linear function in terms of $x$. As far as I know, the most straightforward method is trying to solve the distribution $p(x)$ directly, and then find the first moment by
      $$
      langle xrangle = int x p(x)mathrm dx
      $$
      but sometimes the analytical form of this distribution is very hard to obtain, so instead of doing this is there other method?




      probability differential-equations moment-generating-functions




      share|cite|improve this question













      May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example,
      $$
      fracmathrm d p(x)mathrm dx = alpha(x) p(x)
      $$
      where $alpha(x)$ is a non-linear function in terms of $x$. As far as I know, the most straightforward method is trying to solve the distribution $p(x)$ directly, and then find the first moment by
      $$
      langle xrangle = int x p(x)mathrm dx
      $$
      but sometimes the analytical form of this distribution is very hard to obtain, so instead of doing this is there other method?





      probability differential-equations moment-generating-functions





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 2 at 8:07









      Lorenzo B.

      1,5402318




      1,5402318









      asked Aug 2 at 8:00









      Kawai Cheung

      112




      112

























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