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

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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

2-3-3-cie-fig1-unif-dist

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

fraction numerator 1 over denominator b minus a end fraction

  • Therefore the probability density function is given by

    straight f left parenthesis x right parenthesis equals open curly brackets table row cell fraction numerator 1 over denominator b minus a end fraction end cell cell a less or equal than x less or equal than b end cell row cell space space space space 0 end cell otherwise end table close

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

   Error converting from MathML to accessible text.

  • This is the (vertical) axis of symmetry of the rectangle
  • Should the above be forgotten, begin mathsize 16px style E left parenthesis X right parenthesis equals integral subscript a superscript b x straight f left parenthesis x right parenthesis space straight d x end style 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

   begin mathsize 16px style bold Var bold left parenthesis bold italic X bold right parenthesis bold equals bold 1 over bold 12 begin bold style stretchy left parenthesis b minus a stretchy right parenthesis end style to the power of bold 2 end style

  • Should the above be forgotten, begin mathsize 16px style Var left parenthesis X right parenthesis equals integral subscript negative infinity end subscript superscript infinity x squared straight f left parenthesis x right parenthesis space straight d x minus open square brackets E open parentheses X close parentheses close square brackets squared end style or begin mathsize 16px style V a r left parenthesis X right parenthesis equals E left parenthesis X squared right parenthesis minus open square brackets E open parentheses X close parentheses close square brackets squared end style 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

Worked example

A continuous random variable, X , is modelled by the uniform distribution such that
 straight f left parenthesis x right parenthesis equals 0.4 for a less or equal than x less or equal than 4 and straight f left parenthesis x right parenthesis space equals 0  otherwise.
 a is a constant.

(a)
Show that the value of a is 1.5 .

 

(b)
Find
(i)
straight P left parenthesis 2.5 less or equal than X less or equal than 3 right parenthesis
(ii)
E left parenthesis X right parenthesis
 
(c)
Find the standard deviation of X, giving your answer in the form a square root of 3, where a is a rational number.
(a)
Show that the value of a is 1.5 .

2-3-3-cie-fig2-we-solution_a

(b)
Find
(i)
straight P left parenthesis 2.5 less or equal than X less or equal than 3 right parenthesis
(ii)
E left parenthesis X right parenthesis
2-3-3-cie-fig2-we-solution_b
(c)
Find the standard deviation of X, giving your answer in the form a square root of 3, where a is a rational number.
2-3-3-cie-fig2-we-solution_c

Examiner Tip

  • A sketch of the graph of a uniform distribution is quick and will highlight the symmetry in a uniform distribution
  • Use areas of rectangles to find probabilities rather than integrating

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.