We had mentioned in earlier modules why we need a different way to handle continuous probability spaces and random variables. In continuous spaces, we use pdfs to describe random variables, but pdfs are not probabilities (directly). Rather they are used under an integration sign to get probabilities. Because of this technical development, a couple of allied technical concepts: cummulative distribution functions and “derived” pdfs become helpful (we could have defined them for discrete random variables as well, but they would not have been worth the effort).

We revisit pdfs again, and learn about a new kind of random variable, the exponential distribution. In a sense the exponential is a continuous analog of the geometric distribution, and shares with it what is called the “memoryless” property. It is very closely connected with Poisson random variables as well. We learnt in the last module that the number of chewing gum stains per tile on the sidewalk is a Poisson process (approximately). Assuming the sidewalk were one-dimensional (so only length, no width, the tiles are segments on a line), the distance between stains is an exponential random variable.