Probability density functions and total area | Year 12

01 / LEARN

The idea, step by step

Probability is area

A continuous random variable can take any real value in an interval. Its density describes how probability is spread. A density height is not a probability: an interval probability is the area beneath the curve over that interval.

A density must be nonnegative on its support and have total area one. Outside the stated support it is zero. Including or excluding individual boundary points does not change the area.

f (x) \geq 0, \int_{- \infty}^{\infty} f (x) d x = 1

Normalise, then check

When a model contains an unknown constant, first make its total area equal to one. Then check nonnegativity throughout the support. A total integral of one cannot repair a negative part of the proposed density.

A density can be taller than one: a rectangle of height 2 and width 0.5 has area one. It is probabilities, not density heights, that must stay between zero and one. A single exact value has probability zero.

A common trap

Rejecting any density height above one.

Instead: Check nonnegativity and total area, not an upper bound on height.

02 / WORKED EXAMPLES

See the thinking

Example 1

Find the scale factor

Find k if $f (x) = k x$ on [0,2] is a density.

$\int_{0}^{2} k x d x = 2 k = 1$ .
Thus $k = 1 / 2$ .
The resulting $x / 2$ is nonnegative on [0,2].

k = 1 / 2

Example 2

A tall density

Is $f (x) = 2$ on [0,0.5], zero elsewhere, a density?

It is nonnegative.
Total area is $2 (0.5) = 1$ .
A height greater than one is allowed.

Yes.

Check your understanding

Find c if $f (x) = c$ on [1,5] is a density.

Show answer

$c = 1 / 4$ .

Width 4 gives $4 c = 1$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Is $f (x) = 0.8$ on [0,2] a density? Explain.
2 marks
Find k if $f (x) = k x$ on [0,3] is a density.
2 marks
Check both conditions for $f (x) = 3 x^{2}$ on [0,1].
2 marks
For any continuous density, find $P (X = 0.7)$ .
2 marks
A nonnegative triangular density has width 1 and height 2. Is its area correct?
2 marks
A function has integral one but is negative on part of its support. Is it a density?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Is $f (x) = 0.8$ on [0,2] a density? Explain.
2 marks
Total area $0.8 (2) = 1.6$ , not one.
No.
Find k if $f (x) = k x$ on [0,3] is a density.
2 marks
$\int_{0}^{3} k x d x = 9 k / 2 = 1$ .
$k = 2 / 9$ .
Check both conditions for $f (x) = 3 x^{2}$ on [0,1].
2 marks
$3 x^{2} \geq 0$ and $\int_{0}^{1} 3 x^{2} d x = 1$ .
It is valid.
For any continuous density, find $P (X = 0.7)$ .
2 marks
One point has zero width and zero area.
0.
A nonnegative triangular density has width 1 and height 2. Is its area correct?
2 marks
$\frac{1}{2} (1) (2) = 1$ .
Yes.
A function has integral one but is negative on part of its support. Is it a density?
2 marks
Densities must be nonnegative on their support.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: continuous random variables, normal distributions and sampling distributions of sample means. Original lessons and practice.

Official curriculum source ↗

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