Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Continuous uniform distribution ▷ Mode

Theorem: Let $X$ be a random variable following a continuous uniform distribution:

\[\label{eq:cuni} X \sim \mathcal{U}(a, b) \; .\]

Then, the mode of $X$ is

\[\label{eq:cuni-mode} \mathrm{mode}(X) \in [a,b] \; .\]

Proof: The mode is the value which maximizes the probability density function:

\[\label{eq:mode} \mathrm{mode}(X) = \operatorname*{arg\,max}_x f_X(x) \; .\]

The probability density function of the continuous uniform distribution is:

\[\label{eq:cuni-pdf} f_X(x) = \left\{ \begin{array}{rl} \frac{1}{b-a} \; , & \text{if} \; a \leq x \leq b \\ 0 \; , & \text{otherwise} \; . \end{array} \right.\]

Since the PDF attains its only non-zero value whenever $a \leq x \leq b$,

\[\label{eq:cuni-pdf-max} \operatorname*{max}_x f_X(x) = \frac{1}{b-a} \; ,\]

any value in the interval $[a,b]$ may be considered the mode of $X$.

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Metadata: ID: P84 | shortcut: cuni-med | author: JoramSoch | date: 2020-03-16, 16:29.