Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. distribution function and cumulative distributions function for this discrete uniform Note that \( X \) takes values in \[ S = \{a, a + h, a + 2 h, \ldots, a + (n - 1) h\} \] so that \( S \) has \( n \) elements, starting at \( a \), with step size \( h \), a discrete interval. (probability density function) given by: P (X = x) = 1/ (k+1) for all values of x = 0,... k Note that the mean is the average of the endpoints (and so is the midpoint of the interval \( [a, b] \)) while the variance depends only on the number of points and the step size. The expected value of discrete uniform random variable is E(X)=N+12. Open the Special Distribution Simulator and select the discrete uniform distribution. Recall that \( f(x) = g\left(\frac{x - a}{h}\right) \) for \( x \in S \), where \( g \) is the PDF of \( Z \). Walk through homework problems step-by-step from beginning to end. We specialize further to the case where the finite subset of \( \R \) is a discrete interval, that is, the points are uniformly spaced. For , 2, ..., the first few values are For \( k \in \N \) \[ \E\left(X^k\right) = \frac{1}{n} \sum_{i=1}^n x_i^k \]. Vary the number of points, but keep the default values for the other parameters. and kurtosis excess are, The mean deviation for a uniform distribution on elements is given by. Then \[ H(X) = \E\{-\ln[f(X)]\} = \sum_{x \in S} -\ln\left(\frac{1}{n}\right) \frac{1}{n} = -\ln\left(\frac{1}{n}\right) = \ln(n) \]. Hints help you try the next step on your own. Note that the last point is \( b = a + (n - 1) h \), so we can clearly also parameterize the distribution by the endpoints \( a \) and \( b \), and the step size \( h \). This follows from the definition of the distribution function: \( F(x) = \P(X \le x) \) for \( x \in \R \). Run the simulation 1000 times and compare the empirical density function to the probability density function. Vary the parameters and note the graph of the distribution function. The distribution function \( F \) of \( X \) is given by. Then \( X = a + h Z \) has the uniform distribution on \( n \) points with location parameter \( a \) and scale parameter \( h \). Of course, the fact that \( \skw(Z) = 0 \) also follows from the symmetry of the distribution. This follows from the definition of the (discrete) probability density function: \( \P(X \in A) = \sum_{x \in A} f(x) \) for \( A \subseteq S \). The discrete uniform distribution is implemented in the Wolfram We will assume that the points are indexed in order, so that \( x_1 \lt x_2 \lt \cdots \lt x_n \). Recall that \( \E(X) = a + h \E(Z) \) and \( \var(X) = h^2 \var(Z) \), so the results follow from the corresponding results for the standard distribution. Note that \( \skw(Z) \to \frac{9}{5} \) as \( n \to \infty \). The probability density function \( f \) of \( X \) is given by \[ f(x) = \frac{1}{\#(S)}, \quad x \in S \]. Recall that \begin{align} \sum_{k=1}^{n-1} k^3 & = \frac{1}{4}(n - 1)^2 n^2 \\ \sum_{k=1}^{n-1} k^4 & = \frac{1}{30} (n - 1) (2 n - 1)(3 n^2 - 3 n - 1) \end{align} Hence \( \E(Z^3) = \frac{1}{4}(n - 1)^2 n \) and \( \E(Z^4) = \frac{1}{30}(n - 1)(2 n - 1)(3 n^2 - 3 n - 1) \). Most classical, combinatorial probability models are … Discrete Uniform Distribution The discrete uniform distribution is also known as the "equally likely outcomes" distribution. Suppose that \( Z \) has the standard discrete uniform distribution on \( n \in \N_+ \) points, and that \( a \in \R \) and \( h \in (0, \infty) \). With this parametrization, the number of points is \( n = 1 + (b - a) / h \). of Integer Sequences. Compute a few values of the distribution function and the quantile function. The uniform distribution on a discrete interval converges to the continuous uniform distribution on the interval with the same endpoints, as the step size decreases to 0. \( G^{-1}(1/4) = \lceil n/4 \rceil - 1 \) is the first quartile. Without some additional structure, not much more can be said about discrete uniform distributions. From MathWorld--A Wolfram Web Resource. A random variable with p.d.f. Vary the parameters and note the graph of the probability density function. The simplest is the uniform distribution. Suppose that \( X \) has the uniform distribution on \( S \). \( G^{-1}(1/2) = \lceil n / 2 \rceil - 1 \) is the median. For odd. https://mathworld.wolfram.com/DiscreteUniformDistribution.html. Then \(Y = c + w X = (c + w a) + (w h) Z\). The quantile function \( G^{-1} \) of \( Z \) is given by \( G^{-1}(p) = \lceil n p \rceil - 1 \) for \( p \in (0, 1] \). For the remainder of this discussion, we assume that \(X\) has the distribution in the definiiton. Recall that \begin{align} \sum_{k=0}^{n-1} k & = \frac{1}{2}n (n - 1) \\ \sum_{k=0}^{n-1} k^2 & = \frac{1}{6} n (n - 1) (2 n - 1) \end{align} Hence \( \E(Z) = \frac{1}{2}(n - 1) \) and \( \E(Z^2) = \frac{1}{6}(n - 1)(2 n - 1) \). Of course, the results in the previous subsection apply with \( x_i = i - 1 \) and \( i \in \{1, 2, \ldots, n\} \). [ "article:topic", "showtoc:no", "license:ccby", "authorname:ksiegrist" ], \(\newcommand{\R}{\mathbb{R}}\) \(\newcommand{\N}{\mathbb{N}}\) \(\newcommand{\Z}{\mathbb{Z}}\) \(\newcommand{\E}{\mathbb{E}}\) \(\newcommand{\P}{\mathbb{P}}\) \(\newcommand{\var}{\text{var}}\) \(\newcommand{\sd}{\text{sd}}\) \(\newcommand{\cov}{\text{cov}}\) \(\newcommand{\cor}{\text{cor}}\) \(\newcommand{\skw}{\text{skew}}\) \(\newcommand{\kur}{\text{kurt}}\), 5.21: The Uniform Distribution on an Interval, Uniform Distributions on Finite Subsets of \( \R \), Uniform Distributions on Discrete Intervals, probability generating function of \( Z \), \( F(x) = \frac{k}{n} \) for \( x_k \le x \lt x_{k+1}\) and \( k \in \{1, 2, \ldots n - 1 \} \), \( \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 \). Note the graph of the distribution function. of them having the same probability, then, Restricting the set to the set of positive If \(c \in \R\) and \(w \in (0, \infty)\) then \(Y = c + w X\) has the discrete uniform distribution on \(n\) points with location parameter \(c + w a\) and scale parameter \(w h\).