4.01 Random variables and probability distributions

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En provenance du cours de University of Amsterdam

Basic Statistics

1922 notes

University of Amsterdam

Basic Statistics

1922 notes

Cours 3 sur 5 dans 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.

À partir de la leçon

Probability Distributions

Probability distributions form the core of many statistical calculations. They are used as mathematical models to represent some random phenomenon and subsequently answer statistical questions about that phenomenon. This module starts by explaining the basic properties of a probability distribution, highlighting how it quantifies a random variable and also pointing out how it differs between discrete and continuous random variables. Subsequently the cumulative probability distribution is introduced and its properties and usage are explained as well. In a next lecture it is shown how a random variable with its associated probability distribution can be characterized by statistics like a mean and variance, just like observational data. The effects of changing random variables by multiplication or addition on these statistics are explained as well.The lecture thereafter introduces the normal distribution, starting by explaining its functional form and some general properties. Next, the basic usage of the normal distribution to calculate probabilities is explained. And in a final lecture the binomial distribution, an important probability distribution for discrete data, is introduced and further explained. By the end of this module you have covered quite some ground and have a solid basis to answer the most frequently encountered statistical questions. Importantly, the fundamental knowledge about probability distributions that is presented here will also provide a solid basis to learn about inferential statistics in the next modules.

4.01 Random variables and probability distributions6:39

4.02 Cumulative probability distributions5:20

Rencontrer les enseignants

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

A random variable can in fact be a lot less random than its name suggests.

In this video, I'll explain how a probability distribution

describes the possible outcomes of a random variable with their probabilities.

In that way, the probability distribution makes a lot of the randomness concrete and

paves the road to use a random variable in calculations.

When making observations on individuals or

objects, you can observe several attributes per individual.

These are called variables.

Now imagine you've collected a dataset and you decide to repeat your study.

You could perhaps find the same individuals or objects and

measure the variables again.

Or otherwise you could take a sample and measure similar individuals again.

In any case you will find values for your variables that are different.

Even if you were to measure, for example, a single person's length several times,

the result would most likely deviate a few millimeters up to a centimeter,

depending on time of day, the accuracy of your measuring tape, etc.

So very often you expect

that there is random variation associated with the values of the variable.

If this aspect of randomness due to chance is relevent,

such a variable is called a random variable.

A random variable can take on a set of possible values.

Each with an associated probability.

So if you take a large enough sample for

this random variable, the relative frequencies for

the different values it can take will be equal to the probabilities.

To keep things clear in writing, an italic capital letter is used to denote

a random variable, and a small letter is used to indicate the value that it takes.

There are two types of random variables, discrete and continuous.

A discrete random variable is one which may take on only a countable

number of distinct values, such as 0/1/2/3.

In fact, if a random variable can take on

only a finite number of distinct values then it must be discrete.

Examples of discrete random variables include the number of children in a family

or whether you had to wait in front of a traffic light today.

A continuous random variable is one which

takes an infinite number of possible values.

Continuous random variables are usually measurements.

To illustrate the aspect of infinity, lets assume a height

that has been measured as 3.1 meters, but

with a more accurate measuring tape, a value of 3.14 is measured.

With an even more accurate tape, 3.145 meters.

In other words, by making more accurate measurements,

or zooming in, an infinite number of outcomes is possible.

Other examples of continuous variables are age, temperature, or

the time it would take you to run a mile.

The value that the random value may take

can conveniently be described by a probability distribution.

A probability distribution can take the shape of a table, a graph, or

a mathematical equation, and is defined as a list of probabilities

associated with each of the values that a random variable may take.

Every random variable has by definition a probability distribution.

The probability distribution of a discrete random variable

is called a probability mass function.

While for a continuous variable it is called a probability density function.

I will explain the reason for this distinction in a moment.

For discrete random variables,

it is easy to see how the probability can be listed for every possible outcome.

Suppose a variable X can take the values 1, 2, 3, or 4.

Then the following table lists the probabilities with each outcome.

This distribution may also be described by a probability histogram.

That's lovely, exactly the same as what you would do with the frequency table and

a frequency histogram.

An example of a continuous probability distribution for

the random variable X could be this graph.

This probability distribution does not give probability on the y-axis,

but rather a unit that is called probability density.

The probability per unit value of the x variable.

To get a probability you need to consider a certain interval under the curve

rather than the height of the curve at a certain location.

the probability is then given by the surface area.

The consequence of having a probability density on the y-axis

is that when units of the random variable change, for

example, if you express a length, not in meters, but in centimeters,

the values along the y-axis change accordingly.

In the end the surface area for the same interval should not change.

Let me summarize what I hope you understood from this video.

A random variable is a variable whose possible values are numerical outcomes

of a random phenomenon.

A random variable is discrete if it can take

only a countable number of distinct values, and

it is continuous if it can take an infinite number of possible values.

A probability distribution specifies the probabilities for

each of the values that the random variable may take.

A probability distribution of a discrete random variable is called a probability

mass function and gives probabilities on the y-axis.

A probability distribution for a continuous random variable is called

a probability density function and it gives probability densities on the y-axis.

In this case, probabilities are given by the surface area under the curve

within a specified interval.

A probability density function can exist in the form of a table, a graph,

and an equation.

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