Maths › Further Statistics 2 › Continuous random variables: density and distribution functions
Continuous random variables: density and distribution functions
When a variable can take any value in a range, no single value has a probability. Areas do instead, and the function whose areas they are is the density.
Builds on The normal distribution and Definite integrals and areas.
IN THIS TOPIC
- Check that a proposed density is valid and use it to find probabilities.
- Move between the density and the cumulative distribution function in both directions.
- Use the distribution function to find the median and other percentiles.
COMMON MISCONCEPTION
For a continuous random variable, P(X = 1.5) is the height of the density at 1.5.
Areas, not heights
A probability density function f(x) is never a probability. It is non-negative, its total area is 1, and probabilities are the areas beneath it:
What the booklet prints is the cumulative version, F(x₀) = P(X ≤ x₀) as the integral of f from the bottom of the range up to x₀. This two-limit form is not on the page, so hold on to the idea that probability is area. The area over a single point is zero, so P(X = c) = 0 for every c: the height f(c) is not a probability. A density may even exceed 1, as it does on a short interval, and nothing is wrong when it does. One useful consequence: the inequality signs stop mattering for a continuous variable, which is a genuine simplification after all the care discrete distributions demand.
WORKED EXAMPLE
Checking a density and using it
f(x) = 3x²/8 for 0 ≤ x ≤ 2 and zero elsewhere. Verify it is a density and find P(1 < X ≤ 1.5).
It is non-negative on the interval. Its integral is [x³/8] from 0 to 2 = 8/8 = 1, so it is valid.
P(1 < X ≤ 1.5) = [x³/8] from 1 to 1.5 = 3.375/8 − 1/8 = 0.297 to three decimal places.
The distribution function
The cumulative distribution function collects everything to the left, so F(x0) = P(X ≤ x0), the integral of f from the lower end up to x0. It climbs from 0 to 1 and never falls. Differentiating undoes the integration:
The booklet defines F as an integral of f but never states the reverse, so learn that differentiating the distribution function returns the density. F is the better tool for reading probabilities off, since P(a < X ≤ b) is just F(b) − F(a). f is the better tool for describing shape. Percentiles come from F, with the median m solving F(m) = 0.5 and the lower quartile solving F(q) = 0.25. Piecewise definitions need every branch stated, including the 0 below the range and the 1 above it.
GUIDED PRACTICE
From density to median
For f(x) = 3x²/8 on 0 ≤ x ≤ 2, write down F(x) in full and find the median.
Show the working
Integrating from 0: F(x) = x³/8 on the interval. In full, F(x) = 0 for x < 0, x³/8 for 0 ≤ x ≤ 2, and 1 for x > 2.
The median solves m³/8 = 0.5, so m³ = 4 and m = 1.587 to three decimal places.
Checking: F(1.587) = 3.998/8 ≈ 0.5, and the value sits inside the range as it must.
ASSESSMENT FOCUS
- Verify a density by checking both conditions, non-negative and total area exactly 1.
- Write a piecewise F(x) with every branch, including the flat 0 and the flat 1.
- Read probabilities from F by subtraction. Going back to the integral wastes time you will want later.
- For a percentile, set F equal to the proportion and solve, then check the root lies in range.
- When f is defined in pieces, build F piece by piece and make the branches agree where they meet.
CHECK YOURSELF
f(x) = kx for 0 ≤ x ≤ 4 and zero elsewhere. Find k and F(x) on the interval.
Show a hint
Total area 1 fixes k, then integrate from the lower end.
Show the answer
The area is k[x²/2] from 0 to 4 = 8k, so k = 1/8. Then F(x) = x²/16 on 0 ≤ x ≤ 4, with F = 0 below and F = 1 above.
For a continuous variable, probability is area under the density: P(a < X ≤ b) is the integral of f from a to b, and P(X = c) = 0.
F is the integral of f from the lower end, climbing from 0 to 1; differentiating F gives f back, and F(m) = 0.5 gives the median.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the continuous random variables: density and distribution functions questions page.
CHECK YOUR PROGRESS
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- Check that a proposed density is valid and use it to find probabilities.
- Move between the density and the cumulative distribution function in both directions.
- Use the distribution function to find the median and other percentiles.
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