3.10 Independence between random events

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Из курса от партнера University of Amsterdam

Basic Statistics

2057 оценки

University of Amsterdam

Basic Statistics

2057 оценки

Курс 3 из 5 — Specialization Methods and Statistics in Social Sciences

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

Из урока

Probability

This module introduces concepts from probability theory and the rules for calculating with probabilities. This is not only useful for answering various kinds of applied statistical questions but also to understand the statistical analyses that will be introduced in subsequent modules. We start by describing randomness, and explain how random events surround us. Next, we provide an intuitive definition of probability through an example and relate this to the concepts of events, sample space and random trials. A graphical tool to understand these concepts is introduced here as well, the tree-diagram.Thereafter a number of concepts from set theory are explained and related to probability calculations. Here the relation is made to tree-diagrams again, as well as contingency tables. We end with a lesson where conditional probabilities, independence and Bayes rule are explained. All in all, this is quite a theoretical module on a topic that is not always easy to grasp. That's why we have included as many intuitive examples as possible.

3.08 Joint and marginal probabilities6:02

3.09 Conditional probability4:49

3.10 Independence between random events5:13

3.11 More conditional probability, decision trees and Bayes' Law8:12

Познакомьтесь с преподавателями

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

In this video I'll explain a probability concept that you will

encounter frequently, independence.

If random events are independent, it means that knowing the outcome of one event

does not provide information about the others.

It's an important bit of information because if random events are independent,

it simplifies the probability calculations enormously, and

it's therefore an assumption that's often made.

Let me introduce an example.

You've made a count of different activities by the people on a beach

which you distinguish by gender.

And you've turned it into a table with probabilities.

Joint probabilities in a central block and marginal probabilities here and there.

Independence between random events means that knowing the outcome of one event

does not influence your knowledge about the outcome of the others.

In the example, it means that if you know about the gender of a person,

it would not influence the probabilities for that person's activities.

Or, if you would consider a certain activity,

it would not influence the probabilities for gender.

Would that be reasonable?

Let's have a look.

Compared to probability of selecting a male person from the people who

are resting,

with the probability of selecting a male when disregarding activity.

It turns out that the two probabilities are not quite the same.

We can take a different approach as well to look at independence.

If you could assume independence between two variables, gender and

activity, you could calculate the joint probabilities by

multiplying the marginal probabilities for gender and activity.

Let's see whether this would be reasonable in our case.

We multiply the marginal probability for gender with those for the activities.

If gender and

activity at the beach will be independent, you would see these probabilities.

However, we did observe during probabilities

that they're quite different.

Take a look.

So the assumption of independence seems not to hold.

Here we have applied two ways to check for independence among random variables.

And these two ways are in fact quite close to the formal conditions for independence.

The first is events A and B are independent if the probability of event

A given any outcome of event B is equal to the marginal probability of A.

And in this definition, you could switch the labels A and B.

The second definition is events are independent if

their joint probability equals the product of their separate marginal probabilities.

If any of the two conditions holds, then events A and

B can be considered as independent.

For many real life situations,

independence among random events is not that common at all.

Quite often, there are unexpected and unknown factors

that influence the events of interest so that the specific outcome for

one event turns out to be related to another in a probabilistic sense.

It does not have to be any direct or

any understood relation between events to be dependent in terms of probabilities.

Another aspect to bear in mind when thinking about independence is that,

while the definitions are clear, it is in practice not always that easy

to find sufficient support for the assumption of independence.

You might need a large random sample to gather enough evidence.

And as a final note it is important to be aware

of the difference between disjoint events versus independent events.

If events are disjoint, it means that if one event occurs the others cannot occur.

So by definition, their joint probability is zero.

And as a consequence, disjoint events are very dependent.

Once one event occurs, you know that the probability of the others is zero.

Contrastingly, if events are independent, the occurrence of one event

does nothing about another, and can also not be disjoint.

I hope you've learned the following from this video.

Random events are independent if the joint probability of the events are equal to

the product of the marginal probabilities, which is equivalent to the situation

where the probability of an outcome for one event given any outcome for

any other event is equal to the marginal probability for that event.

For many real life situations,

independence among random events is not that common

because unknown factors may influence the various events at the same time.

And finally you shouldn't confuse the term disjoint event and independent event.

If two events are disjoint, it implies that they are dependent,

whereas if two events are independent, it implies that they cannot be disjoint.

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