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Discrete Uniform Distribution#

Definition: Discrete Uniform Distribution

We say that a [[Random Variables#Discrete Random Variables|discrete random variable]] \(X\) has a uniform distribution if its [[Random Variables|support]] \(S = \{x_1,\dotsc, x_n\}\) is finite and its [[Random Variables|probability mass function]] \(p\) is given by

\[ p(x) = \begin{cases}\frac{1}{n} \qquad x \in S \\ 0 \qquad \text{otherwise} \end{cases} \]

Notation

\[ X \sim U(n) \qquad X \in U(n) \]

Continuous Uniform Distribution#

Definition: Continuous Uniform Distribution

We say that a [[Random Variables#Continuous Random Variable|continuous random variable]] has a uniform distribution on the interval \([a;b]\) (or \((a;b)\)) if its [[Random Variables#Cumulative Distribution Functions|cumulative distribution function]] is given by

\[ F(x) = \begin{cases}0\qquad\text{ if } x \lt a \\ \frac{x-a}{b-a} \hphantom{..} \text{ if } a \le x \le b \\ 1 \qquad \text{ if } x \gt b\end{cases} \]

Notation

\[ X \sim U(a,b) \qquad X \in U(a,b) \]

Properties#

Theorem: Probability Density of Uniform Distributions

The [[Random Variables|probability density function]] of a [[Random Variables#Continuous Random Variable|continuous random variable]] which has the [[Uniform Distribution#Continuous Uniform Distribution|uniform distribution]] \(U(a,b)\) is given by

\[ f(x) = \begin{cases}\frac{1}{b-a} \hphantom{..}\text{ if } a \lt x \lt b \\ 0 \qquad \text{ otherwise }\end{cases} \]
Proof

TODO