State the formula for E(X) for a continuous random variable.

E(X)=\int x\,f(x)\,dx over the range of X.

State the formula for Var(X).

Var(X)=E(X^2)-[E(X)]^2, with E(X^2)=\int x^2f(x)\,dx.

Is E\left(\frac1X\right)=\frac{1}{E(X)}?

No. In general E(g(X))\ne g(E(X)); integrate \frac1xf(x).

How do you find the mode of a continuous random variable?

Find where f(x) is greatest: solve f'(x)=0 and check the end points of the range.

How do you find the median?

Solve F(m)=0.5, i.e. \int{lower}^mf(x)\,dx=\frac12.

When is a distribution positively skewed?

Mean median mode: a long tail to the right.

When is a distribution negatively skewed?

Mean < median < mode: a long tail to the left.

What is the skewness of a symmetrical distribution?

Zero skew: mean, median and mode are equal.

How do you judge whether a continuous model is suitable?

Compare its mean, variance and range with the data, then give a conclusion in context.

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

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State the formula for $\mathrm{E}(X)$ for a continuous random variable.

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State the formula for $\mathrm{E}(X)$ for a continuous random variable.: $\mathrm{E}(X)=\int x\,f(x)\,dx$ over the range of $X$ .
State the formula for $\mathrm{Var}(X)$ .: $\mathrm{Var}(X)=\mathrm{E}(X^2)-[\mathrm{E}(X)]^2$ , with $\mathrm{E}(X^2)=\int x^2f(x)\,dx$ .
How do you find $\mathrm{E}(g(X))$ ?: $\mathrm{E}(g(X))=\int g(x)f(x)\,dx$ .
Is $\mathrm{E}\left(\frac1X\right)=\frac{1}{\mathrm{E}(X)}$ ?: No. In general $\mathrm{E}(g(X))\ne g(\mathrm{E}(X))$ ; integrate $\frac1xf(x)$ .
Give $\mathrm{E}(aX+b)$ and $\mathrm{Var}(aX+b)$ .: $a\mathrm{E}(X)+b$ and $a^2\mathrm{Var}(X)$ .
How do you find the mode of a continuous random variable?: Find where $f(x)$ is greatest: solve $f'(x)=0$ and check the end points of the range.
How do you find the median?: Solve $F(m)=0.5$ , i.e. $\int_{\text{lower}}^mf(x)\,dx=\frac12$ .
How do you find the $p$ th percentile?: Solve $F(q)=\frac{p}{100}$ .
When is a distribution positively skewed?: Mean $>$ median $>$ mode: a long tail to the right.
When is a distribution negatively skewed?: Mean $<$ median $<$ mode: a long tail to the left.
What is the skewness of a symmetrical distribution?: Zero skew: mean, median and mode are equal.
How do you judge whether a continuous model is suitable?: Compare its mean, variance and range with the data, then give a conclusion in context.

Exam questions on Mean and variance of continuous random variables

The continuous random variable $X$ has probability density function $f(x)=\frac{3}{8}x^2$ for $0\le x\le 2$ , and $f(x)=0$ otherwise.
Find $\mathrm{E}\!\left(\frac{1}{X}\right)$ .2 marks
The continuous random variable $Y$ has probability density function $f(y)=\frac{y}{2}$ for $0\le y\le 2$ , and $f(y)=0$ otherwise.
Calculate $\mathrm{E}(Y)$ . Using this value, together with the mode and median of $Y$ , describe the skewness of the distribution, justifying your answer.2 marks
The continuous random variable $X$ has probability density function $f(x)=\frac{3}{x^4}$ for $x\ge 1$ , and $f(x)=0$ otherwise.
Find the 90th percentile of $X$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).