Random Variables and Their Distributions

Lesson 1 of 2: Defining and Tabulating a Random Variable

In this lesson:

• Turn outcomes into the one number you care about
• Build a probability distribution table
• Check that the probabilities account for everything

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Define a random variable by assigning a number to each outcome
2. List the possible values of a random variable and find each probability
3. Build a probability distribution table and verify the probabilities sum to 1

Two Coins: How Many Heads?

Toss two fair coins. The outcomes are HH, HT, TH, TT.

If you're betting on how many heads, you only care about a number: 0, 1, or 2.

Four outcomes — but only three numbers matter. Which would you bet on?

One Die: Read the Number Off

Roll one fair die. Let = the number showing.

Here the outcome is a number — just reads it off:

A Random Variable Reports a Number

A random variable is a rule that assigns a numerical value to each outcome in a sample space.

• Name the variable with a capital letter, usually
• It is really a function from outcomes to numbers
• reports the number, not the outcome itself

Mapping the Two Coins to Values

= number of heads. Every outcome maps to a number:

Notation: Capital X, Lowercase x

• — the variable (the rule itself)
• — a particular value it can take
• reads "the probability that equals 1"

Same letter, two jobs: the rule and one of its values.

Your Turn: Name a Random Variable

A spinner has regions worth 1, 2, 5, and 10 points. You spin once.

Define a sensible and list its possible values.

Commit to an answer before advancing.

From Values to How Likely Each Is

You can name the value. Now: how likely is each value?

• Each of HH, HT, TH, TT is equally likely: probability
• Group the outcomes by their -value, then add

Build the Table by Grouping

= number of heads, each outcome probability :

The Merge: Two Outcomes, One Value

• HT and TH both give , so their probabilities add
• , not

Validity Check: Do They Sum to 1?

• Every possible result is accounted for
• If the sum isn't 1, you probably missed a merge

Your Turn: Girls in Three Children

A family has three children. = number of girls. The 8 outcomes are equally likely.

Values are 0, 1, 2, 3. Find each probability.

How many of the 8 outcomes give each value?

Three Children: Complete and Check

• Probabilities:
• Check: ✓

Two Traps to Watch Out For

The outcome is not the value: — HH is the outcome, 2 is the number

Don't forget the merge: , not — add every way it can happen

Sum must be 1: if it isn't, a merge was missed

What You Built In This Lesson

✓ A random variable turns "what happened" into the number you care about

✓ Its distribution table lists each value with its probability

✓ The sum-to-1 check confirms the table is complete

Coming Up Next: The Balance Point

This distribution has a single balance point — one number summarizing where it all sits.

Next lesson, we graph the distribution and learn to read its shape — then later, we compute that balance point.