Continuous Distribution

A continuous uniform distribution is a ‘continuous’ distribution:

• Any value, [latex]x[/latex], between the lower [latex]a[/latex] and upper limit [latex]b[/latex] is possible
• It differs from ‘discrete’ distributions where only whole numbers are possible for [latex]x[/latex].
• See the graphs of uniform distributions with different min and max values below.

Equal Likelihood

• Uniform means “remaining the same at all times“
• We see from the above graph that the height, [latex]h[/latex], remains the same over each uniform distribution.
• This is due to the fact that there is equal likelihood of each value, [latex]x[/latex], occurring
• This gives each distribution the shape of a rectangle.
• Because of this and the fact that the total area of any probability distribution must equal to 1:
  [latex](b-a) \times h = 1[/latex] or, [latex]h = \frac{1}{b-a}[/latex]
• This gives an area (or probability) between two [latex]x[/latex]-values, [latex]x_1[/latex] and [latex]x_2[/latex]:
  [latex]P(x_1 \le x \le x_2) = (x_2 - x_1) \times \frac{1}{b-a} = \frac{x_2 - x_1}{b-a}[/latex]

Statistical Properties

The following metrics apply to uniform distributions:

• They have a lower limit (lowest possible value): min = [latex]a[/latex]
• They have an upper limit (highest possible value): max = [latex]b[/latex]
• The mean is: [latex]\mu = \frac{a+b}{2}[/latex]
• The standard deviation is:  [latex]\sigma = \frac{b-a}{\sqrt{12}}[/latex]
• The variance is: [latex]\sigma^2 = \frac{(b-a)^2}{12}[/latex]
• The distribution is symmetric, so skewness = 0.

Video Explaining Uniform Distributions

Key Takeaways (EXERCISE)

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