Discrete Uniform Distribution

A discrete random variable has a discrete uniform distribution if each value of the random variable is equally likely and the values of the random variable are uniformly distributed throughout some specified interval.

In this article, I will walk you through discrete uniform distribution and proof related to discrete uniform. You can find solved numerical on discrete uniform distribution examples.

Discrete Uniform Distribution Definition

A discrete random variable $X$ is said to have a uniform distribution if its probability mass function (pmf) is given by

$$ \begin{aligned} P(X=x)&=\frac{1}{N},;; x=1,2, \cdots, N. \end{aligned} $$

Discrete Uniform Distribution Graph

Following graph shows the probability mass function of discrete uniform distribution $U(1,6)$.

Mean of Discrete Uniform Distribution

How do you find mean of discrete uniform distribution? is given below with proof

The expected value of discrete uniform random variable is $E(X) =\dfrac{N+1}{2}$.

Proof

The expected value of discrete uniform random variable is

$$ \begin{aligned} E(X) &= \sum_{x=1}^N x\cdot P(X=x)\\ &= \frac{1}{N}\sum_{x=1}^N x\\ &= \frac{1}{N}(1+2+\cdots + N)\\ &= \frac{1}{N}\times \frac{N(N+1)}{2}\\ &= \frac{N+1}{2}. \end{aligned} $$

Hence, the mean of discrete uniform distribution is $E(X) =\dfrac{N+1}{2}$.

Variance of Discrete Uniform Distribution

The variance of discrete uniform random variable is $V(X) = \dfrac{N^2-1}{12}$.

Proof

The variance of discrete uniform distribution for random variable $X$ is given by

$$ \begin{equation*} V(X) = E(X^2) - [E(X)]^2. \end{equation*} $$

Let us find the expected value of $X^2$.

$$ \begin{eqnarray*} E(X^2) &=& \sum_{x=1}^N x^2\cdot P(X=x)\\ &=& \frac{1}{N}\sum_{x=1}^N x^2\\ &=& \frac{1}{N}(1^2+2^2+\cdots + N^2)\\ &=& \frac{1}{N}\times \frac{N(N+1)(2N+1)}{6}\\ &=& \frac{(N+1)(2N+1)}{6}. \end{eqnarray*} $$

Now, the variance of $X$ is

$$ \begin{eqnarray*} V(X) & = & E(X^2) - [E(X)]^2\\ &=& \frac{(N+1)(2N+1)}{6}- \bigg(\frac{N+1}{2}\bigg)^2\\ &=& \frac{N+1}{2}\bigg[\frac{2N+1}{3}-\frac{N+1}{2} \bigg]\\ &=& \frac{N+1}{2}\bigg[\frac{4N+2-3N-3}{6}\bigg]\\ &=& \frac{N+1}{2}\bigg[\frac{N-1}{6}\bigg]\\ &=& \frac{N^2-1}{12}. \end{eqnarray*} $$

Thus the variance of discrete uniform distribution is $\sigma^2 =\dfrac{N^2-1}{12}$.

The discrete uniform distribution standard deviation is $\sigma =\sqrt{\dfrac{N^2-1}{12}}$.

Discrete uniform distribution Moment generating function (MGF)

MGF of discrete uniform distribution is given by The MGF of $X$ is $M_X(t) = \dfrac{e^t (1 - e^{tN})}{N (1 - e^t)}$.

Proof

Discrete uniform distribution moment generating function proof is given as below

The moment generating function (MGF) of random variable $X$ is

$$ \begin{eqnarray*} M(t) &=& E(e^{tx})\\ &=& \sum_{x=1}^N e^{tx} \dfrac{1}{N} \\ &=& \dfrac{1}{N} \sum_{x=1}^N (e^t)^x \\ &=& \dfrac{1}{N} e^t \dfrac{1-e^{tN}}{1-e^t} \\ &=& \dfrac{e^t (1 - e^{tN})}{N (1 - e^t)}. \end{eqnarray*} $$

General discrete uniform distribution

A general discrete uniform distribution has a probability mass function

$$ \begin{aligned} P(X=x)&=\frac{1}{b-a+1},;; x=a,a+1,a+2, \cdots, b. \end{aligned} $$

Mean of General discrete uniform distribution

The expected value of above discrete uniform randome variable is $E(X) =\dfrac{a+b}{2}$.

Variance of General discrete uniform distribution

The variance of above discrete uniform random variable is $V(X) = \dfrac{(b-a+1)^2-1}{12}$.

Distribution Function of General discrete uniform distribution

The distribution function of general discrete uniform distribution is

$F(x) = P(X\leq x)=\frac{x-a+1}{b-a+1}; a\leq x\leq b$.

Hope you like article on Discrete Uniform Distribution. You can refer below recommended articles for discrete uniform distribution calculator and step by step guide on discrete uniform distribution examples.

  1. Discrete Uniform Distribution Calculator
  2. Discrete Uniform Distribution Examples

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