I know that the PDF is the first derivative of the CDF for a continuous random variable, and the difference for a discrete random variable. However, I would like to know why this is, why are there two different cases for discrete and continuous?

**Answer**

I am gonna be a bit imprecise, but hopefully intuitive.

Discrete and continuous probability distributions must be treated differently. For any value in a discrete distribution there is a finite probability. With a fair coin, the probability of heads is 0.5, with a fair six sided die, the probability of a 1 is one sixth, etc. However, the probability of any specific value in a continuous distribution is zero, because one specific value is only one value out of an infinite number of possible values, and if specific values had a >0 probability, then they would not sum up to 1. Hence, with continuous distributions we talk about the probability of *ranges* of values.

“Sum up to” is key in answering your question. If you are at all familiar with calculus and its history, you understand that the integral sign—that elongated ‘S’: $\int$—is a special kind of summation: one describing the limiting case as we approach summing an infinite number of vanishingly small values between points $a$ and $b$ on some function. If that function is a PDF, we can integrate it (sum up) to produce a CDF, and conversely differentiate (difference) the CDF to obtain the PDF.

In the discrete case, we can simply perform standard arithmetic summation (hence, big ‘$\Sigma$’, rather than the tall ‘S’ notation) and arithmetic differencing.

**Attribution***Source : Link , Question Author : BadBlock , Answer Author : Zhubarb*