Statistics Introduction to Probability Distributions

Hi and welcome to www.educator .com.0000

Today we are going to be talking about probability distributions.0002

We are just going to start talking about them.0006

So far we have covered sampling method as well as a little bit about basics of fundamental of probability.0012

In probability distribution we are going to be playing those two ideas together and work on that so we can get what we want out of them.0019

Because of that we need to talk about what sampling and probability cannot do.0030

They cannot tell us something and because of that we need probability distribution and0035

that is going to be the solution to what sampling and probability alone cannot accomplish.0040

With the probability distribution we are going to be talking about random variable and what that means and the expected value is.0047

We know a little bit about sampling and a little bit about probability, basically in sampling we know that0061

what we are trying to do is estimate some variable by taking the sample of the population.0065

Here is the population of all the people we are interested in and we take a sample out and0071

we look at those individuals to try and estimate what that variable is like in the population.0079

We know a little bit about the probability now in a particular population, we know the chance of getting a specific sample.0092

What is the probability of getting this particular sample.0105

You might want to ask yourself what is the relationship between these two things.0111

It is where we are getting the sample and we are looking at the mean and the standard deviation of some particular variable in order to estimate that variable.0116

In probability, what we are doing is it is not that we are actually taking a sample but we have a population.0130

We are trying to think about all the different kinds of samples, the sample space and then calculate the probability of the specific sample.0139

In one we are actually taking the sample and the other we are not actually taking a sample.0153

Here there are some limitations because even if we try to take a random sample, one that is not biased in some way.0159

Even though we try to take a sample we do not necessarily know how close that sample is to the population.0169

Here we have taken a sample but we know the probability of getting that particular sample.0176

We are still missing one piece of the puzzle, neither sampling nor probability helps us know the actual characteristics,0186

Things like shape and spread of the population.0194

We can know how likely this sample is given the population.0199

We could take our sample out and summarize that sample.0203

But we do not know how to understand the population if we do not already know what it is.0209

What do we do?0216

Here is where the probability distributions are going to come in.0219

In probability distributions, what we are going to do is take this population and look at the distribution of all the different possible samples and get their probabilities.0232

This way it is not that we are calculating one probability for one particular sample but we are calculating all the possible sample and all the probability that go along with it.0259

That is the probability distribution.0271

We are no longer dealing with single probability or single samples but we are looking at the universe of all the sample0274

and the probability that go with them that might happen from a low population.0282

It is not necessary to know or we definitely know that this is a population but it could be a theorize population like a fair sighted point.0292

Those could be theoretical model.0309

That is the known population, we have sense of the population and from that we win from it,0311

all the possible sample and all the possibilities that go with the sample.0321

Here is why we want to do that.0329

First we use probability, the fundamental principles of probability to figure out what samples from real populations look like.0332

We would want to do it from one sample at a time now we are going to do it for all the possible samples.0341

We figure out what the distribution of samples looks like then we take a sample from a different unknown population.0347

Here we have known population and now we get all samples and corresponding probability of those samples.0357

Then we have this unknown one, unknown population, and we take a sample.0377

What we do is we compare this sample to the results of the samples from the known population.0387

Why this sample like this is our theorize population?0394

Why is this sample unlikely in this universe of all possible samples from such a know population.0399

If this sample is highly like given that we know that this is probability then we could say these two guys are similar.0407

But if this sample is very unlikely then maybe we could say perhaps these two population are not similar.0417

This one is different from this known population.0428

That is the insight, we are going to use these probability distributions in order to help us figure out samples0432

that come from population to try and figure out what the unknown population is like.0440

It helps us to find what a probability distribution is.0450

These are all the samples that can be drawn from an a known population and their corresponding probabilities.0456

When you think about probability distributions, I also want you to think of what it is called a sampling distribution.0489

What it is really is, it is not a distribution of a single entity or people or companies.0502

It is not a distribution of single data point.0511

Instead it is a sampling distribution.0515

It is a distribution made up of a whole bunch of samples.0519

Now that we know what probability distributions are, for and why we need them so desperately.0529

Let us try to smoothen out the road from probability to probability distributions.0535

In probability we learn about sample spaces.0546

Remember when we are talking about sample spaces, we are talking about all the different possible samples that we could have.0550

When we talk about flipping two coins and we look at what is the probability of getting 2 heads in a row?0557

We created these sample spaces of all the possible different outcomes.0565

For instance, if we look at the sample space for 2 dice we might look at something like this, dice 1, 2.0579

We might have 1, 2, 3, 4, 5, 6 and 1, 2, 3, 4, 5.0597

If we look at the sample space that would be 1-1, 1-2, 1-3, 1-4, 1-5, 1-6.0610

Or it might be 2-6, 3-6, 4-6, 5-6, 6-6.0622

If you fill in all the different possible outcomes there are 36 possible outcomes.0632

If you fill in the rest of this table you could see the entire sample space.0642

Each of these roll of a dice presuming that this dice are fair, each of these we have a probability of 1 out of 36.0647

The likelihood of this out of all the possible outcomes is 36.0659

That would be looking at the sample space.0668

That is the probabilities of all the different outcomes.0672

When we talk about probability distributions, often we are talking about just one number and the distribution around that one variable, whatever that variable is.0677

Here notice that we have 2 numbers but we may want something like the sum of 2 dice.0690

Here the sum would be 2 and the sum would be 3 and the sum would be 4.0699

Maybe we want the probability distribution that looks something like this.0706

Here I will put all the different possible sums.0713

I may not have room for this so I might draw them separately.0717

For instance, can I have a sum of 0?0720

No, I cannot have a sum of 1 either.0724

If I roll 2 dice the lowest possible sum I could have is 2, a 1 and a 1.0727

I could have a sum of 2, 3, 4, 5, 6, 7 but I could also have, I will continue this on this side.0733

It will also have sums of 8, 9, 10, 11, 12.0747