Uniform Distribution

• Notation
  • $X ∼ Unif(a, b)$

• Interpretation
  • Uniform Distribution is a symmetric probability distribution on a fixed interval. It has a constant probability for any value within the interval ${a,b}$ and 0 elsewhere.

• Type:
  • Discrete or Continuous

• Parameter(s):
  • $a$ - the minimum value allowed
  • $b$ - the maximum value allowed

• Probability Density Function:

  • Discrete Uniform Distribution: {% math %}f(x) = \frac{r}{b - a + 1}{% endmath %}
  • Continuous Uniform Distribution: {% math %}f(x) = \frac{1}{b - a + 1}{% endmath %}

• Range:
  • $x \in (a, b)$

• Mean:

  • Discrete Uniform Distribution: {% math %}E(X) = \frac{1}{b - a + 1}{% endmath %}
  • Continuous Uniform Distribution: {% math %}E(X) =\frac{1}{b - a}{% endmath %}

• Variance:
  • Discrete Uniform Distribution: $Var(X) = \frac{[(b - a + 1)^2 - 1]}{12}$
  • Continuous Uniform Distribution: $Var(X) = \frac{[(b - a)^2]}{12}$

• Application:

  • One of the most important applications of the uniform distribution is in the randomly generation of numbers. The standard uniform distribution can be used for generating random numbers on the {% math %}(0, 1){% endmath %} interval. Furthermore, we can apply transformations to the uniform random variable to generate random numbers in arbitrary range.

  • The uniform distribution defines equal probability over a given range for either a continuous or a discrete distribution, it is often used as a inference distribution in bayesian statistics when we know nothing about our target distribution.