Cumulative and Probability density functions. How do go from one to another

Question

I have a question regarding CDFs and PDFs.

Let $F_Y=P(Y<k)$ be the CDF.I understand that I can simply differentiate the function to find the PDF, $F_y=P(Y=k)$. This means I can integrate and differentiate to go from one to another.

However, I' don't see how given I know the CDF $P_Y$ I can find the pdf $P_y$ of an integer only statistic.

For example, a counting proccess

$\sum_{k=0}^K P(Y=k)=P(Y\leq K)$, but how can I get $P(Y=k)$ given I know $P(Y\leq K)$

Answer 1

If the random variable $Y$ takes only integer values, then $$ \mathbb{P}(Y=k)=\mathbb{P}(Y\leq k)-\mathbb{P}(Y\leq k-1)$$

Note that for a discrete random variable, the usual terminology is probability mass function rather than probability density function.

Answer 2

The relationship, for discrete variates, of the (discrete) probability function (sometimes called probability mass) and the analog of the CDF, is the same as for continuous variates, but replasing integration with summation (and differentiation with first differences.

You have to be meticulous about whether the endpoint is included in the sum or not (it is if the CDF-analogue is $\Bbb{P}(Y \leq k)$) and whether to use the first difference from below or from above.