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In my probability textbook I got stuck on the following problem: cosntruct a positive (it means $\geq 0$) real function $f(x)$ (not necessarily continuous, of course) such that it is constantly $0$ for every $x \leq 0$ and $$ 1 = \int_{- \infty}^{x} f(t) \ \! \mathrm{d}t $$ for all $x >0.$ This problem should be obvious (according to the authors), yet I am not sure it is even true. Any suggestions? Thanks a lot.

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What you want is the Dirac delta "function". (It isn't really a function; it's a distribution.)

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The conditions you specify give the cumulative density function $F(x) = \mathbb{1}_{[0,\infty)}(x)$ which is indeed a valid cumulative density function ($\lim_{x\to -\infty} F(x) = 0, \lim_{x \to \infty} F(x) = 1, \lim_{x \to a^+} F(x)= F(a)$).

This distribution is a point mass at $0$, which as the other answer points out has a "density" as the Dirac delta distribution.

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    $\begingroup$ Let me suggest to stop being shy and to squarely state that there is no such function $f$. This is the most important message to bring to the OP. $\endgroup$ – Did Oct 12 '14 at 14:46
  • $\begingroup$ Ok, that goes with my suspicion. Thank you guys! $\endgroup$ – Jorge.Squared Oct 12 '14 at 15:22

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