Continuous Uniform Distribution (Edexcel International A Level (IAL) Maths: Statistics 2): Revision Note

Continuous Uniform Distribution

What is meant by the continuous uniform distribution?

• This is a special case of a probability density function for a continuous random variable

  • The normal distribution is another special case covered in S1

• The uniform, or rectangular, distribution is a p.d.f. that is constant and non-zero over a range of values but zero everywhere else

• Since the area under the graph has to total 1, the height of the uniform distribution would be

$\frac{1}{b-a}$

• Therefore the probability density function is given by

  $\mathrm{f}\left(x\right)=\{\begin{array}{ll}\frac{1}{b-a}& a\le x\le b\\ 0& \mathrm{otherwise}\end{array}$

How do I find probabilities for a continuous uniform distribution?

• Sketch the graph of y= f(x) 

• Probabilities are the area under the graph, all such areas will now be rectangles

  • Finding the area of a rectangle is likely to be easier than integration!

• The symmetrical properties of rectangles may also be used to find probabilities

How do I find the mean, median, mode and variance of a continuous uniform distribution?

• The mean, or expected value, is given by

   $\mathbf{E}\mathbf{\left(}\mathit{X}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}$

• This is the (vertical) axis of symmetry of the rectangle

• Should the above be forgotten, $E\left(X\right)={\int }_a^{b}x\mathrm{f}\left(x\right)\mathrm{d}x$ can still be applied

  • You be may asked to use this to prove the result

• The median can also be found by symmetry and will be equal to the mean

• There is no mode as f(x) is equal - and so at its greatest - for all values of x

• The variance is given by

   $\mathbf{Var}\mathbf{\left(}\mathit{X}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{12}}{\left(b-a\right)}^{\mathbf{2}}$

• Should the above be forgotten, $\mathrm{Var}\left(X\right)={\int }_{-\infty }^{\infty }{x}^2\mathrm{f}\left(x\right)\mathrm{d}x-{\left[E\left(X\right)\right]}^2$ or $Var\left(X\right)=E\left(X^2\right)-{\left[E\left(X\right)\right]}^2$ can still be applied

  • You may be asked to use this to prove the result

  • The standard deviation is the square root of the variance