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

f(x)\geq0 for all x, and the total area is 1: \int{-\infty}^{\infty}f(x)\,dx=1.

Define the cdf F(x0).

P(X\leq x0)=\int{-\infty}^{x0}f(x)\,dx

For a continuous random variable, what is P(X=a)?

0, so < and \leq give the same probability.

What are F(x) values below and above the range of X?

0 below the range and 1 above it.

How do you find k in a pdf?

Set the total integral of f(x) over its range equal to 1 and solve.

How do you handle a piecewise pdf when finding a probability?

Split the integral at each boundary and use the correct expression in each part.

How does a histogram relate to the pdf?

With narrow classes and frequency density on the vertical axis, the histogram approaches the pdf, with total area 1.

Probability density and cumulative distribution functionsEdexcel A-Level Further Maths: Flashcards

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State the two conditions for a pdf $\mathrm{f}(x)$.

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State the two conditions for a pdf $\mathrm{f}(x)$ .: $\mathrm{f}(x)\geq0$ for all $x$ , and the total area is 1: $\int_{-\infty}^{\infty}\mathrm{f}(x)\,\mathrm{d}x=1$ .
How is $\mathrm{P}(a<X\leq b)$ found from the pdf?: $\int_a^b\mathrm{f}(x)\,\mathrm{d}x$
Define the cdf $\mathrm{F}(x_0)$ .: $\mathrm{P}(X\leq x_0)=\int_{-\infty}^{x_0}\mathrm{f}(x)\,\mathrm{d}x$
How is $\mathrm{f}(x)$ found from $\mathrm{F}(x)$ ?: $\mathrm{f}(x)=\dfrac{\mathrm{d}\mathrm{F}(x)}{\mathrm{d}x}$
$\mathrm{P}(a<X\leq b)$ in terms of $\mathrm{F}$ ?: $\mathrm{F}(b)-\mathrm{F}(a)$
$\mathrm{P}(X>a)$ in terms of $\mathrm{F}$ ?: $1-\mathrm{F}(a)$
For a continuous random variable, what is $\mathrm{P}(X=a)$ ?: $0$ , so $<$ and $\leq$ give the same probability.
What are $\mathrm{F}(x)$ values below and above the range of $X$ ?: $0$ below the range and $1$ above it.
How do you find $k$ in a pdf?: Set the total integral of $\mathrm{f}(x)$ over its range equal to 1 and solve.
How do you handle a piecewise pdf when finding a probability?: Split the integral at each boundary and use the correct expression in each part.
How do you find the value $q$ with $\mathrm{P}(X\leq q)=0.75$ ?: Solve $\mathrm{F}(q)=0.75$ .
How does a histogram relate to the pdf?: With narrow classes and frequency density on the vertical axis, the histogram approaches the pdf, with total area 1.
Integrate $kx^n$ ( $n\neq-1$ ).: $\dfrac{kx^{n+1}}{n+1}+c$

Exam questions on Probability density and cumulative distribution functions

The continuous random variable $X$ has probability density function $\mathrm{f}(x)=kx^2$ for $0\leq x\leq3$ , and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.
Find the cumulative distribution function $\mathrm{F}(x)$ for $0\leq x\leq3$ .2 marks
The continuous random variable $X$ has cumulative distribution function $\mathrm{F}(x)=0$ for $x<1$ , $\mathrm{F}(x)=\dfrac{x^2-1}{8}$ for $1\leq x\leq3$ , and $\mathrm{F}(x)=1$ for $x>3$ .
Find $\mathrm{P}(1.5<X\leq2.5)$ .2 marks
The time $X$ , in hours, that a customer waits for a delivery has probability density function $\mathrm{f}(x)=kx$ for $0\leq x\leq2$ , $\mathrm{f}(x)=2k$ for $2<x\leq4$ , and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.
Show that $k=\frac16$ .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).