Relationship between the PF and CDF for discrete and joint random variables

Question

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.

Answer 1

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*}