Skip to content

Salam Academy

Sign in Create account

LearnCambridge9231Current syllabusPaper 4 · Further Probability & StatisticsContinuous random variables

Paper 4 · Module

Continuous random variables

3 subtopics≈ 100 min in total

Start the module: Probability density functions Open graphing calculator

Probability density functions
For a continuous variable, probability lives in area. A probability density function is a curve whose total area is $1$ ; the probability of landing between $a$ and $b$ is the area under the curve between them, found by integration.
≈ 35 min
Mean and variance
Swap the sum for an integral: $E(X)=\int xf(x)\,dx$ and $E(X^2)=\int x^2f(x)\,dx$ , then $\text{Var}(X)=E(X^2)-[E(X)]^2$ exactly as before. For a symmetric pdf, the mean is simply the centre.
≈ 35 min
Median and percentiles
The median cuts the area under the pdf in half: integrate from the lower limit up to $m$ and set the answer equal to $\tfrac12$ . Quartiles and percentiles work the same way with $\tfrac14$ , $\tfrac34$ or any other fraction.
≈ 30 min