Maximum Likelihood for the Exponential Distribution, Clearly Explained!!!

It's raining outside. But I've got a StatQuest inside, yeah. Hello, I'm Josh Starmer and welcome to StatQuest. Today, we're going to be talking about the exponential distribution and its maximum likelihood estimate. We'll start with a brief introduction of the distribution and what it's used for, and then we'll dive into the math and

nitty-gritty of how maximum likelihood is applied to it. So, what is the exponential distribution? It's a statistical distribution that models the time between events. For example, how long will you wait before you get another text message? Or how much time will pass before the next person views this video? Here's what an exponential distribution

looks like. The x-axis is the amount of time between events. The y-axis is scaled so that the total area under the curve equals 1. If we are interested in the probability of an event, like someone viewing this video, happening within 0 to 5 seconds, we solve for the area under the curve from x = 0 to x = 5 seconds.

Here's the equation for an exponential distribution. You plug in some value for x, and out comes a value for y. Lambda is called the rate parameter, and it is proportional to how quickly things happen. In this graph, lambda equals 1, and this models an event happening, like someone watching this video, on average, every

second. Here, lambda equals 2, and this models someone watching the video on average twice every second. Here, lambda equals 0.5. And this models someone watching this video on average once every 2 seconds. The goal of maximum likelihood is, given a set of measurements, to find an optimal value for lambda. So, assume I collected a lot of data