30. Continuous Random Variables | Mathematics Advanced

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

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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