Year 12 Probability: Density functions

For a continuous random variable, probabilities come from areas under a probability density function. The density itself is not a probability at a single point.

The total area under a valid density function over its supported interval must be 1.

What this means

$P (a \leq X \leq b)$ is found by integrating the density from $a$ to $b$ .
A single point has probability 0 for a continuous variable.
The interval where the density is defined matters just as much as the formula itself.

Worked example

Problem: A random variable has density function

f (x) = k x

for

0 \leq x \leq 2

, and

f (x) = 0

otherwise. Find

k

and then find

P (1 \leq X \leq 2)

The total area under the density must be 1: $\int_{0}^{2} k x d x = 1$ .
Integrate: $k {[x^2 / 2]}_{0}^{2} = k (2) = 1$ .
So $k = 1 / 2$ .
Now find the required probability: $P (1 \leq X \leq 2) = \int_{1}^{2} x / 2 d x$ .
Integrate: ${[x^2 / 4]}_{1}^{2} = 4 / 4 - 1 / 4 = 3 / 4$ .

Answer:

k = 1 / 2

and

P (1 \leq X \leq 2) = 3 / 4

Common tasks

Find an unknown constant so the total area is 1.
Calculate a probability over a given interval.
Use the density to find statistics such as the mean.

Worked example 2

Problem: Suppose

f (x) = 2 x

for

0 \leq x \leq 1

. Find

P (X < 0.5)

Use the definition of probability from area: $P (X < 0.5) = \int_{0}^{0.5} 2 x d x$ .
Integrate: ${[x^{2}]}_{0}^{0.5}$ .
Substitute the bounds: ${(0.5)}^{2} - 0 = 0.25$ .

Answer:

P (X < 0.5) = 0.25

Common mistakes

Treating the density value itself as a probability.
Integrating over the wrong interval.
Forgetting to check that the total area under the density is 1.

Revision focus

Probability work becomes far more reliable when you define the object of the question before calculating. Is the problem about ordered outcomes, unordered selections, a conditional event, a discrete random variable, or a continuous density? That decision controls the method. In many exam questions, the major error is not arithmetic. It is using a correct formula for the wrong kind of probability situation.

A second useful habit is to interpret the result immediately after it is found. Decide whether the answer is a count, a probability, an average outcome, or a spread measure. In continuous settings, remember that probabilities come from areas over intervals, not from single points. In discrete settings, remember that the total probability must still add to one. Those interpretation checks are often the fastest way to catch an error before it reaches the final line.

In probability, the final line should always answer the right event, not just display a calculation. After computing, read the question again and confirm that your sample space, condition, or random-variable interpretation still matches what was asked. That last comparison catches a surprising number of near-miss errors.