Introduction to Discrete and Continuous Random Variables

A random variable is a variable whose value is determined by the outcome of a random experiment. At HL, we study both discrete and continuous random variables in depth, including their measures of centre and spread, and how linear transformations affect them.

Discrete Random Variable

A random variable $X$ that can take only a countable number of distinct values (e.g. 0, 1, 2, 3, ...). Each value has an associated probability $P (X = x_{k})$ , and the sum of all probabilities equals 1: $\sum_{k} P (X = x_{k}) = 1$

Continuous Random Variable

A random variable $X$ that can take any value within a given interval (or union of intervals). It is described by a probability density function (PDF) rather than individual probabilities at each point.

Analogy

Think of a discrete random variable like rolling a die — you get one of a fixed set of outcomes. A continuous random variable is more like measuring the exact height of a student — it can take any value in a range, and the probability of any single exact value is zero.

Expected Value (Mean) of a Discrete Random Variable

Before studying variance, we need the expected value — the probability-weighted average of all possible values:

Expected Value of a Discrete Random Variable

$E (X) = \sum_{k} x_{k} P (X = x_{k})$
This is also called the mean and denoted $μ$ .

Example

Example: A discrete random variable $X$ has the following distribution:

$x$	0	1	2	3
$P (X = x)$	0.2	0.3	0.4	0.1

$E (X) = 0 (0.2) + 1 (0.3) + 2 (0.4) + 3 (0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4$

Mode and Median of a Discrete Random Variable

Mode (Discrete)

The value of $x$ with the highest probability $P (X = x)$ . It is the most likely outcome.

Median (Discrete)

The value $m$ such that $P (X \leq m) \geq 0.5$ and $P (X \geq m) \geq 0.5$ . For small distributions, it is found by accumulating probabilities in order until the cumulative total first reaches or exceeds 0.5.

Example

Example: Using the distribution above:

$x$	0	1	2	3
$P (X = x)$	0.2	0.3	0.4	0.1
Cumulative	0.2	0.5	0.9	1.0

Mode: $x = 2$ has the highest probability (0.4), so mode = 2.

Median: The cumulative probability first reaches 0.5 at $x = 1$ , so median = 1.

Variance of Discrete Random Variables

For a discrete random variable $X$ with mean $μ = E (X)$ , the variance measures how spread out the values are around the mean.

Variance of a Discrete Random Variable

$Var (X) = E [(X - μ)^{2}] = \sum_{k} (x_{k} - μ)^{2} P (X = x_{k})$

In practice, the following equivalent formula is far more efficient for calculation:

$Var (X) = E (X^{2}) - [E (X)]^{2}$

where $E (X^{2}) = \sum_{k} x_{k}^{2} P (X = x_{k})$ .

Note

This shortcut formula comes from expanding $E [(X - μ)^{2}] = E [X^{2} - 2 μ X + μ^{2}] = E (X^{2}) - 2 μ E (X) + μ^{2} = E (X^{2}) - μ^{2}$ . It is the form you should use in almost all exam calculations.

Example

Example: A discrete random variable $X$ has the following distribution:

$x$	1	2	3	4
$P (X = x)$	0.1	0.3	0.4	0.2

Step 1: Find $E (X)$
$E (X) = 1 (0.1) + 2 (0.3) + 3 (0.4) + 4 (0.2) = 0.1 + 0.6 + 1.2 + 0.8 = 2.7$

Step 2: Find $E (X^{2})$
$E (X^{2}) = 1^{2} (0.1) + 2^{2} (0.3) + 3^{2} (0.4) + 4^{2} (0.2) = 0.1 + 1.2 + 3.6 + 3.2 = 8.1$

Step 3: Calculate $Var (X)$
$Var (X) = 8.1 - (2.7)^{2} = 8.1 - 7.29 = 0.81$

Standard deviation: $σ = 0.81 = 0.9$

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AHL 4.14—Properties of discrete and continuous random variables

Introduction to Discrete and Continuous Random Variables

Expected Value (Mean) of a Discrete Random Variable

Mode and Median of a Discrete Random Variable

Variance of Discrete Random Variables

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