# Calculating PDF given CDF

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?

“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.