I would like to know whether the following problem is well posed, and whether there is a solution. Let me make it clear that I have no pretentions of rigor here.
Given a continuum random variable $x$, I am looking for the probability distribution $p(x)$ which maximizes
\begin{equation} - \int_{-\infty}^{+ \infty} p(x) \log p(x) \, dx \end{equation} with the constraints \begin{eqnarray} \int_{-\infty}^{+\infty} p(x) \, \theta(x_{\ast}-x)\, dx &=& q,\\ \int_{-\infty}^{+\infty} p(x) \, dx &=& 1, \end{eqnarray} where $x_{\ast}$ and $0< q < 1$ are given numbers, and $\theta(x)$ is the Heaviside step function.
This is analogous to a maximum-entropy problem with a constraint on the cumulative distribution function (CDF) of $p(x)$, and the constraint that $p$ is normalized to unity.
I tried to solve the problem with Lagrange multipliers, but is not clear to me how to do it: I maximize
\begin{equation} \Lambda[p] \equiv - \int_{-\infty}^{+\infty} p(x) \log p(x) \, dx + \lambda \int_{-\infty}^{+\infty}p(x) \left[ \theta(x_{\ast}-x) - q \right]dx + \lambda' \left[ \int_{-\infty}^{+\infty} p(x) \, dx - 1 \right], \end{equation} where $\lambda$ and $\lambda'$ are Lagrange multipliers. I then take the derivative of $\Lambda$ with respect to $p(x)$, set it to zero, and obtain \begin{equation} p(x) = \frac{1}{Z} \exp\left\{ \lambda[\theta(x_{\ast}-x) - q \right]\} \end{equation} which is not normalizable.
Are you aware of where could be the issue? Or whether there is an alternative formulation of this problem which does not have this normalization issue?
Thank you
