State the cdf of U[a, b].

F(x)=\frac{x-a}{b-a} for a\le x\le b; 0 below a and 1 above b.

Why is the pdf of a uniform distribution a rectangle?

The density is constant on [a, b] and zero elsewhere.

How do you find P(c < X < d) for a uniform distribution?

\frac{d-c}{b-a}, the fraction of the interval covered.

X\sim U[0,10]. Find E(X) and Var(X).

5 and \frac{100}{12}=\frac{25}{3}.

How do you derive the mean of a uniform distribution?

Integrate \frac{x}{b-a} from a to b, then use b^2-a^2=(b-a)(b+a).

Given E(X) and Var(X), how do you find a and b?

Solve a+b=2E(X) and (b-a)^2=12\,Var(X), taking b-a0.

Give an example of a quantity modelled by a uniform distribution.

A rounding error, or a waiting time when arrival is equally likely at any moment.

The continuous uniform distributionEdexcel International A Level Further Maths: Flashcards

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State the pdf of the continuous uniform distribution on [a, b].

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State the pdf of the continuous uniform distribution on [a, b].: $f(x)=\frac{1}{b-a}$ for $a\le x\le b$ , and 0 otherwise.
State the cdf of U[a, b].: $F(x)=\frac{x-a}{b-a}$ for $a\le x\le b$ ; 0 below $a$ and 1 above $b$ .
State the mean of U[a, b].: $E(X)=\frac{a+b}{2}$
State the variance of U[a, b].: $\text{Var}(X)=\frac{(b-a)^2}{12}$
Why is the pdf of a uniform distribution a rectangle?: The density is constant on [a, b] and zero elsewhere.
How do you find P(c < X < d) for a uniform distribution?: $\frac{d-c}{b-a}$ , the fraction of the interval covered.
$X\sim U[2,8]$ . Find the density.: $\frac16$
$X\sim U[0,10]$ . Find $E(X)$ and $\text{Var}(X)$ .: $5$ and $\frac{100}{12}=\frac{25}{3}$ .
How do you derive the mean of a uniform distribution?: Integrate $\frac{x}{b-a}$ from $a$ to $b$ , then use $b^2-a^2=(b-a)(b+a)$ .
Which factorisation helps derive the variance?: $b^3-a^3=(b-a)(b^2+ab+a^2)$
Given E(X) and Var(X), how do you find a and b?: Solve $a+b=2E(X)$ and $(b-a)^2=12\,\text{Var}(X)$ , taking $b-a>0$ .
Give an example of a quantity modelled by a uniform distribution.: A rounding error, or a waiting time when arrival is equally likely at any moment.

Exam questions on The continuous uniform distribution

The continuous random variable $X$ is uniformly distributed over the interval $[2,8]$ .
Find $E(X)$ and $\text{Var}(X)$ .2 marks
The length $L$ cm of a rod cut by a machine is modelled by a continuous uniform distribution over the interval $[49,51]$ .
The machine's target length is 50 cm. Find the probability that a rod is within 0.25 cm of the target.2 marks
The continuous random variable $X$ is uniformly distributed over the interval $[a,b]$ , where $b>a$ .
Show that the cumulative distribution function is $F(x)=\frac{x-a}{b-a}$ for $a\le x\le b$ .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).