1.3.3 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