Continuous Random Variables

In short: A continuous random variable takes any value in a range, like a height or a time. Its probabilities are areas under a probability density function $f (x)$ , with total area 1, so $P (a \leq X \leq b) = \int_{a}^{b} f (x) d x$ .

Starting point: Years 9 and 10

Counted and measured data

Discrete data is counted: goals scored, number of siblings. Only certain values are possible.
Continuous data is measured: height, time, mass. Any value in a range is possible.

Histograms for continuous data

Continuous data is grouped into classes such as $150 \leq h < 155$ . A histogram has bars that touch, because the classes have no gaps between them.

Height (cm)	150 to 155	155 to 160	160 to 165	165 to 170
Frequency	4	10	12	4

There are 30 students. The relative frequency of 160 to 165 cm is $\frac{12}{30} = 0.4$ .

Probability as area

If you draw a relative frequency histogram where each bar's area is its relative frequency, the total area of all bars is 1. The chance a student is in a class equals the area of that bar.

Continuous random variables

Measured data

Any value in a range
Probability of one exact value is 0

Density function f(x)

f(x) ≥ 0
Total area 1

Probability

P(a ≤ X ≤ b) = area
Integrate f from a to b

Cumulative F(x)

P(X ≤ x)
Median: F(m) = 0.5

Mean and variance

μ = ∫ x f(x) dx
Var = ∫ x² f(x) dx - μ²

Year 12Probability density functions

A probability density function (PDF) $f (x)$ has two rules:

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

and probabilities are integrals:

$P (a \leq X \leq b) = \int_{a}^{b} f (x) d x$

Because a single point has no width, $P (X = a) = 0$ , so $<$ and $\leq$ give the same answer.

The whole triangle has area 1. The shaded part, P(X ≤ 1), has area ¼.

Example. $f (x) = \frac{x}{2}$ for $0 \leq x \leq 2$ , and 0 elsewhere.

Check: $\int_{0}^{2} \frac{x}{2} d x = [\frac{x ^{2}}{4}]_{0}^{2} = 1$ .

$P (X \leq 1) = [\frac{x ^{2}}{4}]_{0}^{1} = \frac{1}{4}$ .

Cumulative distribution function. $F (x) = P (X \leq x) = \int_{- \infty}^{x} f (t) d t$ . Here $F (x) = \frac{x ^{2}}{4}$ on $[0, 2]$ . The median $m$ solves $F (m) = 0.5$ : $m^{2} = 2$ , so $m = 2 \approx 1.41$ . Also $F^{'} (x) = f (x)$ .

Mean and variance.

$μ = \int x f (x) d x = \int_{0}^{2} \frac{x ^{2}}{2} d x = \frac{4}{3}$

$Var (X) = \int x^{2} f (x) d x - μ^{2} = \int_{0}^{2} \frac{x ^{3}}{2} d x - \frac{16}{9} = 2 - \frac{16}{9} = \frac{2}{9}$

Is it a pdf? Check both rules: $f (x) \geq 0$ everywhere on the domain, and the total area is 1. A function that dips below the axis, such as $sin x$ on $[0, 2 π]$ , cannot be a pdf.

Finding $k$ for other functions. For $f (x) = k sin x$ on $[0, π]$ : $\int_{0}^{π} k sin x d x = 2 k = 1$ , so $k = \frac{1}{2}$ . For $f (x) = k e^{- x}$ on $[0, 2]$ : $k (1 - e^{- 2}) = 1$ .

Mode. The mode is where $f (x)$ is highest. Find it by setting $f^{'} (x) = 0$ (or at an endpoint). For $f (x) = \frac{3}{4} x (2 - x)$ , the peak is at $x = 1$ .

Quartiles. The first quartile $q$ solves $F (q) = 0.25$ and the third solves $F (q) = 0.75$ . Any percentile works the same way: 60% of cases are done by the $t$ with $F (t) = 0.6$ .

A tail probability. For waiting times with $f (t) = \frac{1}{2} e^{- t /2}$ , $P (T > 4) = \int_{4}^{\infty} \frac{1}{2} e^{- t /2} d t = e^{- 2} \approx 0.135$ .

Using a given derivative. Some means need a result like $\frac{d}{d x} (x sin x + cos x) = x cos x$ , which means $\int x cos x d x = x sin x + cos x + C$ .

Quick summary

Discrete data is counted. Continuous data is measured.
For continuous data, probability is the area over an interval. Total area is 1.
A PDF has $f (x) \geq 0$ and total area 1. $P (a \leq X \leq b) = \int_{a}^{b} f (x) d x$ .
$P (X = a) = 0$ for a continuous variable.
$F (x) = P (X \leq x)$ . The median solves $F (m) = 0.5$ . $μ = \int x f (x) d x$ .

Quiz

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Try 3 questions now

Which of these is continuous data?
1. A number of pets
2. B time to run 100 m
3. C shoe size on a label
4. D goals in a match
Show answer
B: time to run 100 m
Time is measured and can take any value in a range.
Which of these is discrete data?
1. A number of cars in a car park
2. B mass of an apple
3. C height of a tree
4. D temperature
Show answer
A: number of cars in a car park
Cars are counted.
In a histogram of continuous data, the bars:
1. A have gaps between them
2. B touch each other
3. C must all be the same height
4. D are drawn sideways
Show answer
B: touch each other
The classes run into each other.