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Statistics LibreTexts

4.5: Uniform Distribution

  • Page ID
    3269
  • Definition\(\PageIndex{1}\)

    A random variable \(X\) has a uniform distribution on interval \([a, b]\), write \(X\sim\text{uniform}[a,b]\), if has pdf given by
    $$f(x) =\left\{\begin{array}{l l}
    \frac{1}{b-a}, & \text{for}\ a\leq x\leq b \\
    0, & \text{otherwise}
    \end{array}\right.\notag$$

    A typical application of the uniform distribution is to model randomly generated numbers. In other words, it provides the probability distribution for a random variable representing a randomly chosen number between numbers \(a\) and \(b\).


    The uniform distribution assigns equal probabilities to intervals of equal lengths, since it is a constant function, on the interval it is non-zero \([a, b]\). This is the continuous analog to equally likely outcomes in the discrete setting.