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Uniform · PDF / PMFResult
Submit the form to compute the density or mass value at your specified point.
Formulas
Continuous uniform:
\[ f(x) = \begin{cases} \dfrac{1}{b-a}, & a \le x \le b \\[4pt] 0, & \text{otherwise} \end{cases} \]
Discrete uniform:
\[ P(X=x) = \begin{cases} \dfrac{1}{m-k+1}, & x \in \{k,\ldots,m\} \\[4pt] 0, & \text{otherwise} \end{cases} \]
FAQs
Density refers to the continuous uniform case \( \mathcal{U}(a,b) \) where values are real numbers. Mass refers to the discrete uniform case over integers where probabilities are assigned to distinct outcomes.
By definition of the uniform distribution, all values in the support are equally likely in the sense of density. The normalization condition forces \( f(x)=1/(b-a) \) across the entire interval.
Including or excluding endpoints does not change probabilities in the continuous case because single points have probability zero. The density value at endpoints equals \( 1/(b-a) \) when the point is considered within the support.
k and m are integers defining the inclusive range of outcomes. The number of outcomes is \( n=m-k+1 \). Each support value has equal mass \( 1/n \).
Precision does not change the true value. It only controls the number of displayed decimal places. Choose a setting that matches your reporting or comparison needs.
This page focuses on the density or mass at a point. For cumulative probabilities, use a CDF calculator specific to the uniform distribution.