Day 31: The Exponential pdf

Today we are talking this guy! The exponential p.d.f and its C.D.F.

Well, it has been a week, don’t even get me started. But if you’re here you don’t want to hear me complain about my week, that isn’t why we come together! Well today let’s do a bit of a dive into the exponential p.d.f. I hope you’ve brushed up, because this is going to get interesting.*

For those who need a refresher or want to start from the beginning (after all we are on day 31!) you should start (about) here. I say about because we are dealing with some pretty intense statistics at this point, so while that post started our formal discussion, other posts built up to it. Anyway, enough of that let’s get to business.

First thing we should talk about is why we looked at the gaussian and now the exponential pdfs. That is actually easy to explain, the pdfs that we will be covering are fundamental to statistics. If you’re into that sort of thing (and if you aren’t why are you here?!) you’ll see papers referring to these pdf a lot and more importantly most (if not all) will make the assumption that you know them. Which is also why a certain writer is memorizing a few, because we have get to .

Now, let’s take a peek at the exponential pdf and see what we are working with:

Notice how we didn’t formally define λ, well there is a catch to that…

This is the form I was taught, the one I’m memorizing and the one I (think) I’ve memorized. The problem is, most of the time (IE google or even the textbook) it is written like this guy…

The two are very much equal to one another, if (and this is a big IF!) we do a substitution and let λ = 1/ λ where 1/ λ is now the mean time before the first event. If we think about it and plug in some numbers say that we have 10 events over an hour, then we have a rate of 1 event every 0.1 hour. So there was a variable swap or reparametrization, involved to go from the way it is written the first time when compared to the way it was written the second time.

Okay, let’s say you believe me and we are ready to start using the function, what does it tell us, or rather what the heck is it good for?! Well, it is good to model the time until a specific event occurs. For example, how long does the average phone call last, or maybe how long before someone walks into a bank. These examples look at the time before a random event occurs, now maybe you are thinking, but we are dealing with continuous variables, yet it seems like this is a discrete event either it happens or it doesn’t happen. You would be correct, but you need to think of the other variable involved, in these example that would be time before it occurs, which is a continuous random variable like the ones we’ve been dealing with.

Other things to notice, λ can take on any value greater than zero, division limits us here. This limitation is also dealt with in the case of x, since we are dealing with time, it doesn’t make sense to look at negative time. Therefore our function is typically said to only be valid between 0 ≤ x. Also, because λ can have any  positive value there are an infinite possible values and therefore an infinite number of possible exponential distributions. Also something of note, in the more common way of writing it λ is the value at x = 0 (if you plug it in you’ll see why), so you automatically know something about how the pdf will look just by knowing what λ is! If you notice the feature image at the top of this post you can see it yourself for different values of λ.

Now, thankfully the C.D.F. of our p.d.f. looks a little (a lot) nicer than our C.D.F. for our gaussian. In fact it looks like this guy:

Piece of cake (well compared to our gaussian!) right? You may have noticed I used the more common notation for the derivation of our C.D.F. that was because I don’t want anyone confused (like I am!).

So next time maybe we can use this to do an example. I find that it’s best if we have examples to show how these things work. Then we can go over some other pdf and CDF. Oh so fun, I know, but hey you keep reading, so I’ll keep writing.

Until next time, don’t stop learning!

*My dear readers, please remember that I make no claim to the accuracy of this information; some of it might be wrong. I’m learning, which is why I’m writing these posts and if you’re reading this then I am assuming you are trying to learn too. My plea to you is this, if you see something that is not correct, or if you want to expand on something, do it. Let’s learn together!!