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

Find where f(x) is greatest: solve f'(x)=0 and check, or test the end points.

If the median is greater than the mean, what does that suggest about the skew?

Negative skew: the tail is on the left.

Why must you check a stationary point when finding the mode?

It could be a minimum or an end value where f is zero, not the maximum.

Mean, variance, mode, median and quartilesEdexcel International A Level Further Maths: Flashcards

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

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State the formula for $E(X)$ of a continuous random variable.: $E(X)=\int xf(x)\,dx$ over the range where $f(x)\ne0$ .
State the formula for $\text{Var}(X)$ .: $\text{Var}(X)=E(X^2)-[E(X)]^2$
How do you find $E(X^2)$ ?: $E(X^2)=\int x^2f(x)\,dx$
How do you find the mode of a continuous random variable?: Find where $f(x)$ is greatest: solve $f'(x)=0$ and check, or test the end points.
How do you find the median?: Solve $F(m)=\frac12$ .
How do you find the lower quartile?: Solve $F(Q_1)=\frac14$ .
How do you find the upper quartile?: Solve $F(Q_3)=\frac34$ .
Define the interquartile range.: $Q_3-Q_1$
$f(x)$ is increasing on $[0,2]$ . Where is the mode?: At $x=2$ , the upper end.
If the median is greater than the mean, what does that suggest about the skew?: Negative skew: the tail is on the left.
$F(y)=\frac{y^2}{4}$ on $[0,2]$ . Find the median.: $\sqrt2$
Why must you check a stationary point when finding the mode?: It could be a minimum or an end value where f is zero, not the maximum.

Exam questions on Mean, variance, mode, median and quartiles

The continuous random variable $X$ has probability density function $f(x)=\frac{3x^2}{8}$ for $0\le x\le2$ , and $f(x)=0$ otherwise.
Find the probability that $X$ is greater than its mean.2 marks
The continuous random variable $Y$ has probability density function $f(y)=\frac y2$ for $0\le y\le2$ , and $f(y)=0$ otherwise.
Write down the mode of $Y$ , giving a reason.2 marks
The continuous random variable $X$ has probability density function $f(x)=12x^2(1-x)$ for $0\le x\le1$ , and $f(x)=0$ otherwise.
Find the mode 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).