Mean and variance of continuous random variablesEdexcel A-Level Further Maths: Revision notes

For a continuous random variable $X$ with probability density function $f(x)$, the expected value (mean) is $\mathrm{E}(X)=\int_{-\infty}^{\infty}x\,f(x)\,dx,$ integrating only over the range where $f(x)\ne0$. The variance is $\mathrm{Var}(X)=\mathrm{E}(X^2)-[\mathrm{E}(X)]^2,\quad\text{where }\mathrm{E}(X^2)=\int x^2f(x)\,dx.$ The standard deviation is $\sqrt{\mathrm{Var}(X)}$. Example: $f(x)=\frac38x^2$ on $0\le x\le2$. $\mathrm{E}(X)=\frac38\int_0^2x^3\,dx=\frac32$, $\mathrm{E}(X^2)=\frac38\int_0^2x^4\,dx=\frac{12}{5}$, so $\mathrm{Var}(X)=\frac{12}{5}-\frac94=\frac3{20}$.

To find the mean of any function of $X$, integrate the function against the density: $\mathrm{E}(g(X))=\int g(x)\,f(x)\,dx.$ This covers $\mathrm{E}(X^2)$, $\mathrm{E}\left(\frac1X\right)$, $\mathrm{E}(\sqrt X)$ and so on. For a linear function, $\mathrm{E}(aX+b)=a\mathrm{E}(X)+b$ and $\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$. Example: with $f(x)=\frac38x^2$ on $[0,2]$, $\mathrm{E}\left(\frac1X\right)=\int_0^2\frac1x\cdot\frac38x^2\,dx=\frac38\left[\frac{x^2}{2}\right]_0^2=\frac34$.

The mode is the value of $x$ where $f(x)$ is greatest. Differentiate $f$ and solve $f'(x)=0$, but also check the end points of the range: for an increasing density such as $f(y)=\frac y2$ on $[0,2]$ the mode is the end point $y=2$. The cumulative distribution function is $F(x)=\int_{-\infty}^xf(t)\,dt$. The median $m$ satisfies $F(m)=0.5$, and the $p$th percentile $q$ satisfies $F(q)=\frac{p}{100}$. The lower and upper quartiles are the 25th and 75th percentiles. Example: $f(x)=3x^{-4}$ for $x\ge1$ gives $F(x)=1-x^{-3}$, so the 90th percentile solves $x^{-3}=0.1$, giving $x=10^{1/3}=2.15$.

Skewness describes the asymmetry of a distribution. Compare the mean, median and mode:

• Positive skew: mean $>$ median $>$ mode (a long tail to the right).
• Negative skew: mean $<$ median $<$ mode (a long tail to the left).
• Zero skew: mean $=$ median $=$ mode, as for a symmetrical density. Always justify the description with the values. Example: $f(y)=\frac y2$ on $[0,2]$ has mean $\frac43$, median $\sqrt2$ and mode $2$, so $\frac43<\sqrt2<2$ and the skew is negative.

A density $f$ can be used to model a real quantity. To assess it, compare the model's mean, variance, median and percentiles with the data's, and check the range. A good model agrees closely with the sample mean and variance and does not rule out values that actually occur. Example: if a model gives mean $3.33$ and variance $5.56$ and a sample has mean $3.4$ and variance $5.8$, the model fits the average behaviour. If $f(d)=0$ for $d>10$ but a value of $12.5$ is observed, the model is unsuitable for the upper tail. Always finish with a judgement in context.