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

Author

Paul

Expertise

Maths

## 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

• Therefore the probability density function is given by

#### 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

• This is the (vertical) axis of symmetry of the rectangle
• Should the above be forgotten,  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

• Should the above be forgotten, or  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,  , is modelled by the uniform distribution such that
for and   otherwise.
a is a constant.

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

(b)
Find
(i)
(ii)

(c)
Find the standard deviation of X, giving your answer in the form , where a is a rational number.
(a)
Show that the value of a is 1.5 .

(b)
Find
(i)
(ii)
(c)
Find the standard deviation of X, giving your answer in the form , where a is a rational number.

#### Exam 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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