How can we "derivate" the P.M.F. (discret random variable) from the C.D.F?

Question

If we can find the C.D.F. by integrating the P.D.F. (or the other way around by derivating the C.D.F). How can we find the Probability Mass Function of a discret variable from the C.D.F.?

I know I'm able to find the C.D.F. from a probability function of a discret variable by finding the general term of the probability summation series (https://en.wikipedia.org/wiki/Series_(mathematics)) as SUM(p(x))=1, inside the range of the original PMF, lets say 2

Thanks.

Kavi Rama Murthy · Accepted Answer · 2019-12-26T05:11:48.503

3

In general $F_X(x)-F_X(x-)=P(X=x)$ $\cdots $ (1)for any random a variable $X$ with CDF $F_X$. So this formula gives pmf for any discrete random a variable $X$. [Here $F_X(x-)$ is defined as $\lim_{y\to x, y<x} F_X(y)$].

In the case of integer valued random variables there is a simpler formula: $P(X=n)=F_X(n)-F_X(n-1)$. A similar formula holds when ever $X$ takes values in a set with no limit points. If $X$ takes all rational values we have to use (1). There is no simpler formula in this case.

edited Dec 26 '19 at 05:11

answered Dec 26 '19 at 00:16

Kavi Rama Murthy

311,013

In that case, I will find the original f(x)=P(X=x) (let's say, a binomial or a poisson distribution) from CDF by looking for the limits and substracting? – Limanido Dec 26 '19 at 01:06
@Limanido I have edited may answer. – Kavi Rama Murthy Dec 26 '19 at 05:12

How can we "derivate" the P.M.F. (discret random variable) from the C.D.F?

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