What does f(a) represent for a continuous random variable?

A density, not a probability. Only areas under f are probabilities.

How do you find an unknown constant k in a pdf?

Integrate f over its range, set the total equal to 1 and solve for k.

Equation that defines the median m?

\int{-\infty}^{m}f(x)\,dx=\frac12

Equation that defines the lower quartile?

\int{-\infty}^{Q1}f(x)\,dx=\frac14

Equation that defines the upper quartile?

\int{-\infty}^{Q3}f(x)\,dx=\frac34

What is f(x) outside the stated range?

0, so only integrate over the range where f is non-zero.

Probability density functionsAQA A-Level Further Maths: Flashcards

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State the two conditions for $f(x)$ to be a pdf.

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State the two conditions for $f(x)$ to be a pdf.: $f(x)\ge0$ for all $x$ , and $\int_{-\infty}^{\infty}f(x)\,dx=1$ .
What does $f(a)$ represent for a continuous random variable?: A density, not a probability. Only areas under $f$ are probabilities.
Value of $P(X=a)$ for a continuous random variable?: $0$
Formula for $P(a<X<b)$ ?: $\int_a^bf(x)\,dx$
Why is $P(X<a)=P(X\le a)$ for a continuous variable?: Because $P(X=a)=0$ .
How do you find an unknown constant $k$ in a pdf?: Integrate $f$ over its range, set the total equal to 1 and solve for $k$ .
Equation that defines the median $m$ ?: $\int_{-\infty}^{m}f(x)\,dx=\frac12$
Equation that defines the lower quartile?: $\int_{-\infty}^{Q_1}f(x)\,dx=\frac14$
Equation that defines the upper quartile?: $\int_{-\infty}^{Q_3}f(x)\,dx=\frac34$
What is $f(x)$ outside the stated range?: $0$ , so only integrate over the range where $f$ is non-zero.
What should you do with a root outside the range of $X$ when solving for a median?: Reject it.
Find $k$ if $f(x)=kx^2$ on $[0,3]$ .: $9k=1$ , so $k=\frac19$ .

Exam questions on Probability density functions

The continuous random variable $X$ has probability density function $f(x)=kx^2$ for $0\le x\le3$ , and $f(x)=0$ otherwise, where $k$ is a constant.
Find $P(1<X<2)$ .2 marks
The continuous random variable $X$ has probability density function $f(x)=\frac{x}{18}$ for $0\le x\le6$ , and $f(x)=0$ otherwise.
Find the median of $X$ .2 marks
The continuous random variable $X$ has probability density function $f(x)=kx(3-x)$ for $0\le x\le3$ , and $f(x)=0$ otherwise, where $k$ is a constant.
Show that $k=\frac29$ .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).