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LearnEdexcel9FM0First teaching 2017Option · Further Statistics 2Continuous probability distributions

Option · Module

Continuous probability distributions

4 subtopics≈ 130 min in total

Start the module: The continuous uniform distribution Open graphing calculator

The continuous uniform distribution
If $X$ is equally likely to be anywhere between $a$ and $b$ , then $X\sim U[a,b]$ . Its pdf is a flat rectangle of height $\dfrac1{b-a}$ , so any probability is just (width of the interval) ÷ (total width). Its mean is the midpoint and its variance is $\dfrac{(b-a)^2}{12}$ .
≈ 30 min
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

Edexcel 9FM0 Continuous probability distributions — A Level Further Mathematics · Salam Academy