Continuous Random Variables (PDF & CDF)
Learn continuous random variables for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-07). Quantities such as times and heights can take any value in a range, so probabilities come from the area under a curve rather than from a list of separate outcomes.
You will learn to test whether a function is a valid probability density function, find probabilities as areas by integration, build the cumulative distribution function and use it to find the median and quartiles, and work with the continuous uniform case — modern statistics in action.
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.