Continuous Random Variables (PDF & CDF)
Continuous random variables are described by a probability density function, where probability is area under the curve, and the cumulative distribution function gives probability up to a value.
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
A continuous random variable uses a probability density function, where probability is area under the curve. This Year 12 Mathematics Advanced topic (MAV-12-07) covers PDFs and CDFs.
A PDF \(f(x)\) has \(f\ge0\) and \(\int f=1\); \(P(a
Method
- Check \(f\ge0\) and \(\int f=1\) (find a constant if needed).
- Probability is the area \(\int_a^b f\,dx\).
- CDF: \(P(a\le X\le b)=F(b)-F(a)\).
| \(P(X<1)\) | \(=\) | \(\tfrac14\times1=0.25\) |
| \(\int_0^{0.5}2x\,dx\) | \(=\) | \(0.25\) |
| \(2k\) | \(=\) | \(1\) |
| \(k\) | \(=\) | \(0.5\) |
| \(F(3)-F(1)\) | \(=\) | \(\tfrac34-\tfrac14=0.5\) |
Common pitfalls
Frequently asked questions
What is a probability density function?
A function f, non-negative with total area 1, where the probability that X lies in an interval is the area under f over that interval.
Why is P(X = a) = 0 for a continuous variable?
Because a single point has no width and hence no area under the density curve.
What is the cumulative distribution function?
F of x equals the probability X is at most x; it is the running area under the PDF.
How are the PDF and CDF related?
The PDF is the derivative of the CDF, and the CDF is the integral of the PDF.