Continuous random variables and distribution functions
Move between a probability density and cumulative distribution, evaluate expectations and transform a continuous variable.
Section Paper 4 · Further Probability & Statistics · Topics 4.1–4.5
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Use piecewise probability density functions and normalisation
- Find probabilities, percentiles and expectations of functions
- Relate PDFs and CDFs and transform a variable in simple cases
A density gives probability through area
A probability density f(x) is non-negative and integrates to 1 over its support. P(a<X<b)=∫ₐᵇ f(x)dx; a single point has probability zero for a continuous variable.
The cumulative distribution F(x)=P(X≤x)=∫_{−∞}ˣ f(t)dt. Where differentiable, f(x)=F′(x). A percentile q solves F(q)=p for the required cumulative probability p.
Expectations and transformations use the distribution
For a function g, E[g(X)]=∫g(x)f(x)dx. This is not generally equal to g(E[X]). Use the given support and split a piecewise density at its breakpoints.
For a monotonic transformation Y=h(X), probabilities can be found from F_Y(y)=P(h(X)≤y), carefully reversing the inequality if h is decreasing. Then differentiate the CDF to obtain the density when needed.
Pause and explain
Which habit best checks your work on continuous random variables and distribution functions?
Worked example
Let X have density f(x)=2x for 0<x<1. Find P(X>1/2) and E[X²].
Show the worked solution
- F(x)=∫₀ˣ2t dt=x² for 0≤x≤1.
- P(X>1/2)=1−F(1/2)=1−1/4=3/4.
- E[X²]=∫₀¹x²(2x)dx=2∫₀¹x³dx=1/2.
Answer P(X>1/2)=3/4 and E[X²]=1/2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A density is f(x)=kx on 0≤x≤2. Find k and then P(X≤1).
Give me a hint
The total area under the density must equal 1.
Compare my reasoning
- 1=∫₀²kx dx=k[x²/2]₀²=2k.
- So k=1/2.
- P(X≤1)=∫₀¹x/2 dx=1/4.
k=1/2 and P(X≤1)=1/4.
Look for these in your work
- Normalisation integral correct
- Constant solved
- Probability integrated over correct interval
A CDF is F(x)=0 for x<0, x²/4 for 0≤x≤2, and 1 for x>2. Find the density on its support and the 75th percentile.
Give me a hint
Differentiate the CDF on the continuous support, then solve F(q)=0.75.
Compare my reasoning
- For 0<x<2, f(x)=F′(x)=x/2.
- Set q²/4=3/4, giving q²=3.
- The percentile lies in the support, so choose q=√3.
f(x)=x/2 for 0<x<2; the 75th percentile is √3.
Look for these in your work
- CDF differentiated
- Cumulative probability set correctly
- Support checked
Let X have CDF F_X and define Y=X³. Explain how to obtain F_Y(y) when X is non-negative.
Give me a hint
Because cubing preserves order for non-negative X, translate Y≤y back to an event about X.
Compare my reasoning
- For y≥0, Y≤y is equivalent to X≤y^{1/3}.
- Therefore F_Y(y)=F_X(y^{1/3}).
- Differentiate this expression where the CDF is differentiable to obtain the density of Y.
F_Y(y)=F_X(y^{1/3}) for y≥0; then differentiate for the density.
Look for these in your work
- Monotonicity used
- Inequality transformed
- CDF-to-density step stated
Common mistakes
- Treating density height as a probability
- Forgetting to normalise a proposed density
- Replacing E[g(X)] with g(E[X])
What can you explain now?
How is P(a<X<b) found from a continuous density f?
Compare with the explanation
Integrate f from a to b
Probability is area under the density over the interval.
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Make it stick
What is the difference between a density, a CDF, a probability and an expectation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.