3.08 Joint and marginal probabilities

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From the course by University of Amsterdam

Basic Statistics

1845 ratings

University of Amsterdam

Basic Statistics

1845 ratings

Course 3 of 5 in the 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.

From the lesson

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

Meet the Instructors

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

Counts of interesting phenomena from everyday life are often turned

into proportions and interpreted as probabilities.

By exploiting the power of probabilities calculus the relations among these

phenomena can be understood better or used to make predictions.

Joint and marginal probabilities are two important types of probabilities

that are encountered in this context.

In this video I'll explain the meaning of joint and

marginal probabilities and show their properties.

You're at the beach and observe your fellow beach visitors.

You notice three different activities among them.

which are mutually exclusive.

People resting, they are all sitting or laying on the beach.

People playing, these are running around, building sand castles or

standing in the water.

And finally, people swimming.

Furthermore, you distinguish by gender.

So each person you observe is a case in your data set, while gender and

activity are variables.

You end up with the following contingency table.

In total you counted 113 people, 79 of whom are resting,

20 are playing, and only 14 of them are swimming.

There are few more female than male visitors, 62 versus 51.

These numbers are in fact the row and

column totals for each variable separately.

These row and column totals are located in the margin of this table and

therefore called marginal values.

Please note that this is no bearing with the colloquial meaning of marginal which

may be unimportant.

In a table like this the marginal values represent account for

a single variable without regarding any other variable.

For example the number of people resting without regarding gender.

Now, we turn this frequency table into a table with proportions

by dividing the number in every cell by the total number, 113.

In this table, the central block contains six positions, which together sum to one.

Also the proportion in each column

add to the values in the marginal row at the bottom and the proportions in each row

add up to the values in the marginal column at the right.

And as a consequence,

the marginal column values add up to one as do the marginal row values.

Your count can be considered as a random sample,

measuring the distribution of activity and gender at this beach,

and you can then think of the proportions as probabilities.

In the central block you have the intersection of activity with gender.

For example, the probability that the given person is male and swimming.

These values are called joint probabilities.

Joint probability is just a short name for

the probability of the intersection of various events.

Each joint probability in our table is a disjoint event from any

other joint probability as each person at the beach

is put only in one of the six joint categories and not counted double.

At the same time, the joint probabilities form a jointly exhaustive set of events

because there is no other outcome for activity and gender possible in this case.

Therefore, the joint probabilities sum to one.

In the margins you have, as you might expect, marginal probabilities.

These give the probabilities when considering only one variable.

For example, the probability that the given person is male regardless of his

activities, or that every person would be playing regardless of gender.

The marginal probabilities result from the union of the joint probabilities, for

example the probabilities of resting, playing, and swimming, per gender.

So here, the addition rule applies.

Probabilities are summed.

Consequently, if you do not have the original counts to your availability, but

only joint probabilities,

you could always calculate the marginal probabilities by summation.

Conversely, if you were given marginal probabilities,

you will not in every case be able to calculate the joint probabilities.

Let me summarize what I have explained in this video.

When you have counts on phenomena that originate from a random sample or

trial, you can turn these into probabilities.

If you have observations on multiple random variables,

you can calculate joint and marginal probabilities for these variables.

Joint probabilities are the probabilities for

the intersection of certain outcomes of the variables, and marginal

probabilities are the probabilities for the outcome of each individual variable.

In a special case, where you have two discrete random variables,

a good way to organize this data is in a contingency table, where you would have

joint probabilities in the center and marginal probabilities at the boundaries.

All the joint probabilities sum to one and

they also add up to the marginal probabilities in both directions.

You can always reconstruct the marginal probabilities from the joint probabilities

by summation.

But reconstructing joint probabilities on the basis of marginal probabilities cannot

be done without making additional assumptions.

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