5. Binomial Distribution & Discrete Random Variables
Intro to Discrete Random Variables
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So one of the very first things we learned how to do when given a dataset was calculate the mean. And, basically, we just added up a bunch of values and divided by the total number of observations to get a number, like 1. Well, now we've been talking a lot about probability distributions, which are a little bit different. Here, we have outcomes of an experiment with their associated probabilities. But nonetheless, one of the things you may have to do is calculate the mean of a random variable. So I wanna show you how to do that in this video. And basically, we're gonna use an equation where we're gonna be adding a bunch of things, but there's one other operation we have to do first. I wanna show you how to do that, and we'll do an example together. Let's get started here. So to find the mean of a discrete random variable, how do we do that? Why can't we just add up a bunch of these numbers and divide by the total number? For example, let's say I had a probability distribution where I would count up the number of kids that are in a random household for a random town. And let's say this was my probability distribution, 0, 1, or 2, with their associated probabilities. So why can't I just add up these numbers, 0, 1, and 2, and divide by 3? Well, it's basically because each of these numbers can't be treated as equals. Each one has their associated probabilities, which are different. For example, 1 kid in a household would be the most common at 60%, whereas 0 kids would be 15%. So you can't treat these things as equals. Alright? So then how do we find the mean? Well, basically, what we do is we're gonna have to do one other thing, which is we're gonna have to multiply each value, in other words, e h x, by its associated probability p of x. So you have to do x times p of x, and then that's when you can add up everything. So as an equation, so if you have to find mu, you basically you're gonna still do a sigma, which is add up everything, but you're gonna have to do x times p of x first. Now, by the way, one of the things that you may see in a course or a book is the expected value. Basically, that means the exact same thing. Whenever you're asked to find the expected value of a distribution, that just means find the mean. So let's go ahead and do that for this specific example here. So one of the things I want you to notice here is that a lot of times the data that you'll be given in these types of problems is gonna list the horizontal is gonna list the outcomes and probabilities horizontally like this. What I want you to do is if you're not given a table like this one, I want you to make one where the outcomes are listed vertically, and then you basically have columns like this for the probability. And then I want you to make one for x times p of x. This makes it much easier to read off your calculations. For example, if I wanna figure out x times p of x, so I can add up everything, I could just go horizontally across 0 times 0.15. Alright? So 0 times 0.15. 0 times anything is just 0. Right? 1 times 0.60. So that's gonna be 1 times 0.60, which is just 0.60. And then finally, I have 2 times 0.25. 2 times 0.25 becomes 0.50. Okay? So now I have all my x times p of x's. Remember, this equation says once you get all the x times p of x's, you have to add up everything. Okay? So basically, just gonna add up everything straight down this column, and that's gonna be my expected value, e x. Alright? So if I just go ahead and add up everything, what I'm gonna notice here is that this is a number 1.1. This would be my mu or e of x. Those things would be they mean the exact same thing. Alright? So that's my answer. And there's 2 things I wanna talk about here. Right? 1 of well, first of all, this is a number, my expected value, that isn't one of my outcomes. It's not 1, or 2, and that's perfectly fine. So you might be asking, how do I get 1.1 kids? Basically, what this number represents, the way that you interpret it, is if I were to sample, let's say, a 1000 or 10000 households, count up all the number of kids, and then take an average or a mean, my average would be around 1.1. That would be about roughly my mean. Alright? So even that's not though it's not one of my outcomes and I can't get 1.1 kids, if I were to do a large enough sample and then take the mean, that would be roughly how many, average kids I have per household. Okay? So that's what that number means. And the second thing I wanna discuss is that this number tends to be close to the number that has the highest probability. Alright? Because, again, if you were to take a bunch of random samples of this, the most likely outcome that you're gonna get is a bunch of ones. And so when you average out all that data and take the mean, it's gonna be a number that's close to this number over here. Alright? So that's it for how to calculate our mean of a discrete random variable. Now that we understand this, let's go ahead and get some practice. Thanks for watching.
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