Relationship between the PF and CDF for discrete and joint random variables I am hoping that someone can explain to me how one function can be obtained from the other for the Probability Function and the Cumulative Distribution Function. 
 A: Let $X$ be a discrete random variable whose possible values are say, $x_{1},x_{2},\cdots,x_{n}$.
The probability function of a discrete random  variable is called probability mass function and is defined as follows:
\begin{eqnarray*}
p(x)&=&P(X=x)
\end{eqnarray*}
where $x$ is an arbitrary value in $\{x_{1},x_{2},\cdots,x_{n}\}$.
The Cumulative Distribution function of a discrete random variable is defined by
\begin{equation*}
F(x_{k})=P(X\leq x_{k})=\sum_{x=x_{1}}^{x_{k}}p(x)
\end{equation*}
From the definitions PMF and CDF, it is easy to see that given $p(x)$, CDF $F(x_{k})$ can be obtained by just adding probabilities of the values $\{x_{1},x_{2},\cdots,x_{k}\}$. 
Given $F(x)$, the $p(x_{k})$ can be obtained by $F(x_{k})-F(x_{k-1})$.
Suppose $X$ is continuous random variable with PDF $f(x)$ and CDF $F(x)$. Given $f(x)$, CDF is obtained from 
\begin{equation*}
F(x)=\int_{\infty}^{x}f(t)dt.
\end{equation*}
Given CDF $F(x)$, the PDF of $X$ can be obtained from
\begin{equation*}
f(x)=\frac{d}{dx}F(x)
\end{equation*}
