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Discrete random variables

Turn the outcomes of an experiment into numbers. Start with what a random variable is and the events it describes, then tell discrete from continuous random variables, build a probability mass function and read its shape, and find probabilities from the cumulative distribution function.

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What is a random variable?

Learn what a random variable is, find its value on each outcome, and use its values to describe events.

X: \Omega \to \mathbb{R}

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Lesson 01 · What is a random variable?

0,0

Ω

H H

H T

T H

T T

0

\frac{1}{2}

1

P

0

1

2

X

Step 1: An outcome

The diagram draws each outcome as an oval inside $Ω$ . An outcome is one element of $Ω$ , one result of the experiment, such as $H T$ .

Step 2: An event

An event is a set of outcomes. The event "exactly one head" is the set ${H T, T H}$ .

Step 3: The probability $P$

$P$ assigns each event a probability, so its one arrow starts from the whole set. For fair coins,

P ({H T, T H}) = \frac{1}{2}

Step 4: The random variable $X$

A random variable $X$ assigns each outcome a number, so an arrow starts from every outcome. Here $X$ counts heads.

Step 5: What is random?

Tossing the coins picks one outcome at random, with chances given by $P$ . The arrows, the rule, never change.

Only the outcome is random. The value of

X

follows from it.

Toss the coins to pick an outcome.

H H

, so

X = 2

H T

, so

X = 1

T H

, so

X = 1

T T

, so

X = 0

The diagram from the lesson itself, running live.

4 of 8 lessons

4 lessons you can take today, and 4 more being written. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

What is a random variable?

Preview

12 questions

Learn what a random variable is, find its value on each outcome, and use its values to describe events.

Covers

What a random variable is

The value of a random variable

Events described by a random variable

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Discrete and continuous random variables

12 questions

Learn to tell whether a random variable is discrete or continuous, and to find the support of a discrete random variable, the set of values it can take.

Covers

Discrete or continuous?

The support

Finding a support from the outcomes

Probability mass functions

12 questions

Learn how to build the probability mass function of a discrete random variable from its outcomes, add its entries to find the probability of any event about the variable, and check the two conditions every PMF meets.

Covers

Building a PMF from the outcomes

Probabilities from a PMF

Checking a PMF is valid

PMFs given by a formula

12 questions

Learn to turn a probability mass function given by a formula into its table and check it is valid, find the unknown constant that makes a formula a PMF, and use it to find probabilities.

Covers

From a formula to a table

Finding an unknown constant in a PMF

Probabilities from a PMF with an unknown constant

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Functions of a random variable

In drafting

—

Bar charts and the shape of a distribution

In drafting

—

The cumulative distribution function

In drafting

—

Probabilities from a CDF

In drafting

Where this unit sits in the curriculum

This unit is part of our Essential Probability & Statistics for ML learning path, which contains 88 lessons. Every one of them is drawn below.

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4 lessons in this unit

30 prerequisite lessons across the unit, counting every step back to the start

54 other lessons in the learning path - after this unit, or alongside it

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Builds on: Exponents and roots, Functions, Inequalities and absolute value, Probability fundamentals

Start Discrete random variables

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More in Probability and Statistics: Sample spaces and events, Counting, Probability fundamentals, Conditional probability and independence, Bayes' theorem and evidence

See it taught

What is a random variable?

Step 1: An outcome

Step 2: An event

Step 3: The probability P

Step 4: The random variable X

Step 5: What is random?

4 of 8 lessons

Where this unit sits in the curriculum

Start Discrete random variables

Step 3: The probability $P$

Step 4: The random variable $X$