The moments of $$X$$ are ordinary arithmetic averages. $$Z$$ has probability generating function $$P$$ given by $$P(1) = 1$$ and $P(t) = \frac{1}{n}\frac{1 - t^n}{1 - t}, \quad t \in \R \setminus \{1\}$. #' @param p vector of probabilities. The probability density function $$g$$ of $$Z$$ is given by $$g(z) = \frac{1}{n}$$ for $$z \in S$$. The chapter on Finite Sampling Models explores a number of such models. distribution are therefore. For selected values of the parameters, run the simulation 1000 times and compare the empirical mean and standard deviation to the true mean and standard deviation. Note the graph of the probability density function. distribution function and cumulative distributions function for this discrete uniform and kurtosis excess are, The mean deviation for a uniform distribution on elements is given by. Suppose that $$Z$$ has the standard discrete uniform distribution on $$n \in \N_+$$ points, and that $$a \in \R$$ and $$h \in (0, \infty)$$. Recall that $$f(x) = g\left(\frac{x - a}{h}\right)$$ for $$x \in S$$, where $$g$$ is the PDF of $$Z$$. 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 $$k \in \{1, 2, \ldots n - 1 \}$$, $$\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$$. $$G^{-1}(3/4) = \lceil 3 n / 4 \rceil - 1$$ is the third quartile. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Compute a few values of the distribution function and the quantile function. From MathWorld--A Wolfram Web Resource. Vary the parameters and note the graph of the distribution function. Then $H(X) = \E\{-\ln[f(X)]\} = \sum_{x \in S} -\ln\left(\frac{1}{n}\right) \frac{1}{n} = -\ln\left(\frac{1}{n}\right) = \ln(n)$. Discrete Uniform Distribution The discrete uniform distribution is also known as the "equally likely outcomes" distribution. The discrete uniform distribution is implemented in the Wolfram #' #' @param x,q vector of quantiles. For the remainder of this discussion, we assume that $$X$$ has the distribution in the definiiton. The quantile function $$G^{-1}$$ of $$Z$$ is given by $$G^{-1}(p) = \lceil n p \rceil - 1$$ for $$p \in (0, 1]$$. The quantile function $$F^{-1}$$ of $$X$$ is given by $$G^{-1}(p) = a + h \left( \lceil n p \rceil - 1 \right)$$ for $$p \in (0, 1]$$. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Most classical, combinatorial probability models are based on underlying discrete uniform distributions. 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. $$X$$ has probability density function $$f$$ given by $$f(x) = \frac{1}{n}$$ for $$x \in S$$. Compute a few values of the distribution function and the quantile function. A random variable $$X$$ taking values in $$S$$ has the uniform distribution on $$S$$ if $\P(X \in A) = \frac{\#(A)}{\#(S)}, \quad A \subseteq S$. Open the Special Distribution Simulation and select the discrete uniform distribution. The discrete uniform distribution is a special case of the general uniform distribution with respect to a measure, in this case counting measure. 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 $$\E(Z^4) = \frac{1}{30}(n - 1)(2 n - 1)(3 n^2 - 3 n - 1)$$. A random variable with p.d.f. Our first result is that the distribution of $$X$$ really is uniform. We specialize further to the case where the finite subset of $$\R$$ is a discrete interval, that is, the points are uniformly spaced. Vary the number of points, but keep the default values for the other parameters. 0, 1/2, 2/3, 1, 6/5, 3/2, 12/7, ... (OEIS A086111 The limiting value is the skewness of the uniform distribution on an interval. Legal. Note the size and location of the mean$$\pm$$standard devation bar. The distribution corresponds to picking an element of $$S$$ at random. There are a number of important types of discrete random variables. Thus $$k = \lceil n p \rceil$$ in this formulation. Open the Special Distribution Simulation and select the discrete uniform distribution. Like all uniform distributions, the discrete uniform distribution on a finite set is characterized by the property of constant density on the set. Open the Special Distribution Simulation and select the discrete uniform distribution.

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