Random Variables Revision Notes for HSC SSCE Mathematics Advanced

Random Variables

What is a random variable?

A random variable is a variable that links the outcomes of a random process to numerical values. When we conduct a statistical experiment or observe a random phenomenon, the random variable assigns a specific number to each possible outcome.

For a discrete random variable, we assign a numerical value to each outcome in the sample space. The sample space is the set of all possible outcomes of the experiment.

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Key notation conventions:

Random variables are represented by capital letters such as $X$ or $Y$
Specific numerical outcomes are shown using lowercase letters, for example $x = 3$

This notation helps us distinguish between the variable itself (which can take different values) and a particular value that the variable takes.

Example: Rolling a die

Consider rolling a fair six-sided die. We can define a random variable $X$ to represent the number that appears on the top face of the die.

The sample space is:

$S = \{1, 2, 3, 4, 5, 6\}$

The random variable $X$ takes the values $1, 2, 3, 4, 5, 6$ .

In this case, $X$ maps each outcome of the die roll directly to its corresponding number.

Worked example 1: Drawing cards

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Worked Example: Drawing Cards from a Deck

An experiment involves drawing a card from a standard deck of $52$ cards.

Part a: Define a random variable $X$ for the number of aces drawn.

Solution:

Let $X$ be the number of aces drawn from the deck.

We need to identify all possible values by examining the sample space. When drawing one card, either an ace is drawn or it is not.

If one card is drawn and it is an ace, then $X = 1$ . If the card drawn is not an ace, then $X = 0$ .

Part b: List the possible values of $X$ .

Solution:

The possible values are:

$X = \{0, 1\}$

This means $0$ for non-ace cards and $1$ for ace cards.

The random variable $X$ assigns numerical values to the outcomes, linking the sample space to numbers.

Discrete vs. continuous random variables

Random variables are classified into two main types based on the nature of their possible values.

Discrete random variables

A discrete random variable is a numerical variable that can only take specific, countable values. These values can be listed out completely, even if the list is very long.

Discrete random variables typically:

Take distinct, countable values such as integers
Often represent things you can count, like the number of students or number of goals

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Example: The number of heads when flipping two coins, where $X = 0, 1, 2$ , is a discrete random variable.

Continuous random variables

A continuous random variable is a numerical variable that can take any value within an interval or range. The values form a continuum and cannot be listed individually.

Continuous random variables typically:

Take any value in an interval, such as real numbers
Often represent measurements like height, time, or temperature

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Example: The time taken to complete a task, measured in minutes as $X$ , is a continuous random variable because time can be any positive real number like $2.5$ or $3.142$ minutes.

Key differences

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The fundamental difference between these two types lies in the nature of their possible values:

Discrete: Countable values that can be listed (often integers). Examples include the number of students in a class or goals scored in a match.
Continuous: Uncountable values in an interval that cannot be listed individually (often measurements). Examples include height, time, or temperature.

Worked example 2: Classification

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Worked Example: Classifying Random Variables

Classify the following scenarios as discrete or continuous random variables, and provide a practical example for each.

Part a: Number of goals scored in a soccer match

Solution:

Let $X$ be the number of goals scored. The possible values are $0, 1, 2, ...$ (countable), so $X$ is a discrete random variable.

Practical example: In a school soccer match, the team could score $3$ goals, meaning $X = 3$ .

Part b: Time taken to solve a puzzle

Solution:

Let $Y$ be the time in minutes to solve the puzzle. The possible values are any positive real number (for example, $2.5$ minutes or $3.142$ minutes), so $Y$ is a continuous random variable.

Practical example: Solving a mathematics puzzle might take $Y = 4.7$ minutes.

When classifying random variables:

Check if the values are countable (discrete) or form an interval (continuous)
Discrete variables often involve counting distinct items
Continuous variables often involve measuring quantities

Remember!

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Key Points to Remember:

A random variable assigns numerical values to the outcomes of a random process, creating a link between the sample space and numbers.
Use capital letters ( $X$ , $Y$ ) for random variables and lowercase letters ( $x = 3$ ) for specific numerical outcomes.
Discrete random variables have countable, distinct values that can be listed, such as the number of goals scored or aces drawn.
Continuous random variables can take any value within an interval and cannot be listed individually, such as time, height, or temperature.
The key difference is in the nature of possible values: countable (discrete) versus any value in an interval (continuous).

Random Variables (HSC SSCE Mathematics Advanced): Revision Notes