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Probability Distribution

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Hi. This tutorial covers the Probability Distribution. So before we actually make a distribution, let's make sure we have some terminology down. OK? So a random variable is a variable whose value is a numerical outcome of a random phenomenon. So if you have a chance experiment, and your data that's being produced are numerical outcomes, we can call that variable a random variable.

OK. Now, let's also distinguish between a discrete random variable and a continuous random variable. A discrete random variable is a random variable that produces data in clearly distinguished integral amounts. So if we think about an experiment of asking the shoe size of the next person to walk through the door, they would be giving you numerical outcomes.

So the data collected would be numerical, and it'd be a random phenomenon, because we won't know who's going to walk through the door next. It's going to be discrete, because shoe sizes are 7, 7 1/2, 8, 8 1/2, 9. OK? There's no way to get an 8.25 as a shoe size or a 9.734. OK? So since they're in clearly distinguished integral amounts, this situation would be a discrete random variable.

A continuous random variable is a random variable that produces data that can be measured as finely as is practical. So if we think of that same experiment, but now instead of asking their shoe size, let's say that we measure their height. OK? In this case, height would be continuous, because we can measure height as finely as is practical. If we had a really good measuring tool, we could get a really, really fine measurement of their height. OK? We are limited by what we're measuring with, but in this case, height is a continuous variable and, in this situation, a continuous random variable.

Now, let's look at a probability distribution. A probability distribution is very similar to a frequency distribution, except instead of frequencies, we're going to have probabilities. All right. Now, a probability distribution of x is a description of all the possible values of a random variable and their probabilities.

All probabilities must be between 0 and 1, and the sum of all probabilities must be 1. OK? So these are important here. So they all have to be between 0 and 1, can ever have in probability less than 0 or greater than 1, and then when you add up all those probabilities, they need to be equal 1, or 100%.

All right. So now, let's introduce a chance experiment that we'll use to make a probability distribution. So our chance experiment is to roll two dice and sum the values shown on both dice. So what I'm going to do is I'm going to shake some dice, and I end up with a 5 and a 2. So the sum there would be 7. OK?

If I rolled it again, I got another sum of 7. OK. If I roll again, I got a sum of 5. OK? So let's define the variable x as the sum of the two rolled dice. OK? So those values that I was saying would be a value of x.

So now, what are the possible values of x? Well, the smallest possible sum on two dice is 1 and 1 which would have a sum of 2. The largest possible value is 6, and really-- or excuse me. It'd be 6 and 6, so the largest possible sum would be 12, and then really, you can have all of the values between 2 and 12. So possible values, 2, 3, 4, 5, dot, dot, dot, 11, 12, so you can have all of those values as possible values of x.

So what type of variable is x? So since there are a countable number of outcomes, this is a discrete random variable. OK? It's not continuous, because there are a countable number of values. OK?

Now, before we can make a probability distribution, we must consider all of the possible outcomes which is known as the sample space of the chance experiment. OK? So what I have laid out here is a table that we're going to be able to look at all of the different values of x, so all of the different outcomes. OK? So here, I have die 1, so the outcomes of die 1, and then here, I have the outcomes of die 2. OK?

So if I'm talking about this outcome here, that's a 1 and a 1, and in the boxes, I'm going to write down the sum. OK? So that outcome would be this one, 1 and 1 has a sum of 2. OK? Now, 1 and 2 would have a sum of 3. OK? That's going to be a different outcome than 2 and 1, because 1 and 2 there is different than 2 and 1 here.

Those are two separate outcomes. OK? So we put a 3 here. OK? 2 and 2 is 4 there, and then I'm just going to go through and fill out the rest of the table here.

All right. So now that we have that filled out, you can see that it seems to be that the most frequently occurring sum was 7. OK? We can see that there are six total ways of getting a 7. OK? Getting a 2 is pretty rare. So snake eyes is getting a 2.

That's pretty rare, and 12 that's also pretty rare. And you can see that some are more common than others. We can also see that there are 36 possible outcomes, so 36 possible outcomes. OK? So if we're calculating theoretical probability of a certain sum, it's always going to be out of 36.

So now, if we want to make our probability distribution, so this is a discrete probability distribution, what we're going to do is have two columns. We're going to have x, and we're going to have the probability of x. OK? So then, what I'm going to put here are the different possible values of x, and now I'm going to write down the probabilities within each the values here. So since there is only one way of getting a 2, this probability is 1 out of 36. OK?

Now, if we can look back at 3, so there were two ways of getting a sum of 3. OK? So this probability ends up being 2/36. OK? Then, going back to 4, 4, there were one, two, three ways of getting a 4. OK? And then looking ahead to 5, there are one, two, three, four ways. OK? So for 4, this would be 3 out of 36. This would be 4 out of 36. OK?

Let's just check 6 and 7. So 6, there are one, two, three, four, five ways of getting a 6, and there are one, two, three, four, five, six ways of getting a 7. And then you can see that 8, 9, 10, 11 mirror what was happening with 2, 3, 4. OK? So let's just fill out the rest of this now, and a lot of times, what we can also do is convert these into decimal values. So we can easily do that, if we wanted to as well.

OK. Now, let's look at a quick continuous probability distribution. So the situation is a bus stops at a particular stop every 10 minutes. After running some errands, a rider arrives at the stop. Let's assume the rider can arrive at any time during the 10-minute interval with equal chance, and we're going to let x represent the number of minutes a bus rider has to wait for the bus. OK. So x is going to represent a continuous random variable, because we could measure the wait time as finely as practical.

OK. Now, so what I have here is I have a table broken up into bins, so we'll have to do some binning. After we make this table, then we can make up the probability distribution. OK. So if we take a look at x, I have them split into basically 1-minute intervals. So if there are 10 of these 1-minute intervals, each probability would end up being 0.1. But what we can do is use this to imagine what the probability distribution will look like. OK?

So the probability distribution is going to be x and p of x. Let's go from 0 out to 10, and what this is going to end up being is a uniform distribution with a height of 0.1. And what that allows us to conclude is that every value between 0 and 10 has an equal probability of being a value of x. All right. This has been your tutorial on the Probability Distribution. Thanks for watching.