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We have the following necessary and sufficient conditions for a pdf/pmf $f_X(x)$:

  1. $f_X(x)\geq 0$ for all $x$
  2. $\sum_x f_X(x)=1$ if pmf or $\int_{-\infty}^\infty f_X(x)dx=1$ if pdf

To prove necessity is simple; follows directly from definitions. The converse should also be simple, since it is relegated in Casella & Berger's Statistical Inference (p. 36 Thm 1.6.5), but I'm having trouble completing the proof. They state that given $f_X(x)$ satisfying those properties we can define $F_X(x)$ and appeal to the necessary and sufficient properties of cdf's (limit 0 at $-\infty$ and 1 at $\infty$, nondecreasing, and right-continuous) and leave it at that. Not really sure how best to define $F_X(x)$ to achieve this, though, nor how those conditions for cdf's play a part in this. Do we have to define it separately for the cases of $f_X(x)$ being a pmf/pdf and then prove the cdf conditions (thus proving a proper cdf associated with the pmf/pdf which, as an earlier theorem shows, fully determines the distribution of some r.v. $X$), or are we instead using the conditions in some way under the assumption that we have a cdf given? Any guidance is appreciated.

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    $\begingroup$ There are (a) distributions which are a mixture of discrete and absolutely continuous distributions so in a sense having a combination of pmf and pdf and (b) distributions which are continuous but not absolutely continuous (such as the Cantor distribution) so having neither a pmf nor a pdf. It seems you are excluding both of these $\endgroup$ – Henry Jun 13 at 22:36
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You can define the distribution function by $$ F_X(x) = \int_{-\infty}^x f(y)dy.$$

In case of a discrete random variable, the distribution function will be

$$ F_X(x) = \sum_{y \leq x} f(y).$$

If you have a pdf/pmf $f_X(x)$ that satisfies conditions 1. and 2., then you can show that your defined $F_X(x)$ will satisfy the conditions that you have already listed as sufficient conditions for $F_X(x)$ to be a proper cdf. Specifically, condition 1. will lead to non-decreasing $F_X(x)$, condition 2. will lead to proper limits, and the right-continuity will be obtained by construction.

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    $\begingroup$ And for a pmf $F_X(x) = \sum\limits_{y \le x} f(y)$ $\endgroup$ – Henry Jun 13 at 22:37
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    $\begingroup$ Thank you for adding this! $\endgroup$ – Misius Jun 14 at 8:00

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