Cumulative distribution functions | Year 12

01 / LEARN

The idea, step by step

Accumulate from the left

The cumulative distribution F(x) is the probability that X is at most x. For a bounded support, F is zero before its lower edge, rises as density area accumulates, and is one beyond its upper edge.

A continuous CDF does not decrease, stays between zero and one, and has no jumps. Flat sections mean there is no density there. The interior formula must not be applied outside the interval where it was obtained.

F (x) = P (X \leq x), P (a < X \leq b) = F (b) - F (a)

Reverse the accumulation

Where a CDF is differentiable, its derivative is the density. This is the same connection between integration and differentiation used in calculus; the cumulative area supplies the constant.

A quantile is found by solving F(q)=p. In particular, the median satisfies F(m)=0.5. A median divides probability into equal halves, not necessarily the physical length of the support. Select a root that lies in the support.

A common trap

Applying an interior CDF formula outside its support.

Instead: Include the outer branches F=0 and F=1.

02 / WORKED EXAMPLES

See the thinking

Example 1

CDF and median

For density $2 x$ on [0,1], find the CDF and median.

Inside the support, $F (x) = \int_{0}^{x} 2 t d t = x^{2}$ .
Use F=0 below zero and F=1 above one.
Solve $m^{2} = 1 / 2$ with $m \geq 0$ .

Interior

F (x) = x^{2}

; median

1 / \sqrt{2}

Example 2

Shifted uniform support

X is uniform on [2,6]. Find F(3) and P(3<X<5).

Interior $F (x) = (x - 2) / 4$ .
$F (3) = 1 / 4$ .
$F (5) - F (3) = 3 / 4 - 1 / 4 = 1 / 2$ .

F (3) = 1 / 4

; interval probability

1 / 2

Check your understanding

If $F (x) = x^{3}$ on [0,1], find the density there.

Show answer

$3 x^{2}$ .

Differentiate the interior CDF.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

For CDF $x^{2}$ on [0,1], find $P (0.4 < X < 0.8)$ .
2 marks
Give the CDF of density $3 x^{2}$ on [0,1] over the whole real line.
3 marks
Find the 90th percentile for CDF $x^{2}$ on [0,1].
2 marks
Can a CDF decrease from 0.6 to 0.5 as x increases? Explain.
2 marks
X is uniform on [4,10]. Find F(2) and F(12).
2 marks
For CDF $1 - {(1 - x)}^{2}$ on [0,1], find the density and median.
3 marks

Read the worked solutions

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

For CDF $x^{2}$ on [0,1], find $P (0.4 < X < 0.8)$ .
2 marks
${0.8}^{2} - {0.4}^{2} = 0.48$ .
0.48.
Give the CDF of density $3 x^{2}$ on [0,1] over the whole real line.
3 marks
Integrate $3 t^{2}$ from 0 to x inside the support; use outer cumulative limits.
0 below 0; $x^{3}$ on [0,1]; 1 above 1.
Find the 90th percentile for CDF $x^{2}$ on [0,1].
2 marks
Solve $q^{2} = 0.9$ in the support.
$\sqrt{0.9} \approx 0.9487$ .
Can a CDF decrease from 0.6 to 0.5 as x increases? Explain.
2 marks
An increasing upper bound cannot remove already accumulated probability.
No.
X is uniform on [4,10]. Find F(2) and F(12).
2 marks
2 is below the support; 12 is above it.
0 and 1.
For CDF $1 - {(1 - x)}^{2}$ on [0,1], find the density and median.
3 marks
Differentiate the CDF.
Solve ${(1 - m)}^{2} = 1 / 2$ and retain the root in [0,1].
$f (x) = 2 (1 - x)$ ; median $1 - 1 / \sqrt{2}$ .

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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.

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