mgf of discrete uniform distribution

Moment Generating Function of Continuous Uniform Distribution The moment generating function is the extreme case of a Then $T$ is a random variable, and since $P(T = n) = r_n$, $r$ is the distribution function for $T$. voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In this case Peter loses on the first trial, regains his initial position in the next $m - 1$ trials, and gains the lead in the next $n - m$ trials. $S_1 = X_1$. I've proven the first part, but I don't seem to know how to take the derivative of this function to find the mean of the distribution. To see this, first note that if $X$ and $Y$ are independent, then $e^{tX}$ and $e^{tY}$ are independent (see Exercise [sec 5.2]. Uniform distribution - Math I should find $M(t) = \frac{ e^{tb} - e^{ta}}{ t(b - a)}$, what am I missing here? distribution, Moment generating function of a degenerate 5.1. Then the moment generating function MX of X is given by: MX(t) = et(1 ent) n(1 et) Proof $$M_X(t)= E(e^{tX})=\sum_{k=1}^n \frac{1}{n}e^{tx_k}.$$ For a fair, six-sided die, there is an equal . Hi Everyone! If X has a discrete uniform distribution f(x) =1/k for x=1,2 - Quora Covalent and Ionic bonds with Semi-metals, Is an athlete's heart rate after exercise greater than a non-athlete. Discrete Uniform Distribution - an overview | ScienceDirect Topics Step 3 - Enter the value of x. It follows from this observation that there is no way to load two dice so that the probability that a given sum will turn up when they are tossed is the same for all sums (i.e., that all outcomes are equally likely). In this case, we have \[g(t) = \sum_{j = 0}^n e^{tj} p(j)\ ,\] and we see that $g(t)$ is a in $e^t$. Note that $h(1) = g(0) = 1$, $h'(1) = g'(0) = \mu_1$, and $h''(1) = g''(0) - g'(0) = \mu_2 - \mu_1$. Thank you so much, just a question on the last step: should it be 1-e^10 and 1-e^t? That is, what is the expected value of $T$? You'll get a detailed solution from a subject matter expert that helps you learn core concepts. . Letting a set have elements, each of them having the same probability, then. \right|_{t = 0} \\ &=& \mu_n\ .\end{aligned}\] It is easy to calculate the moment generating function for simple examples. In short, you use the discrete uniform distribution when you have n possible outcomes that are equally likely to occur. From the definition of a moment generating function : MX(t) = E(etX) = etxfX(x)dx. uniformly distributed with each possible digit,$ 0, 1, 2, , 9$ occurring with probability $\frac{1}{10}.$ Uniform Distribution Moment Generating Function Proof. Then with these choices, we have $E(X) = E(Y) = 7/2$ and $V(X) = V(Y) = 9/4$, and yet certainly $p_X$ and $p_Y$ are quite different density functions. How long will it take? Moment generating function for the uniform distribution From $-\infty$ to $a$, for example, you are integrating $(0)e^{tx}$. f( x) = \left\{ \begin{array}{l l} In probability theory, a symmetric probability distribution that contains a countable number of values that are observed equally likely where every value has an equal probability 1 / n is termed a discrete uniform distribution. Both the moment generating function $g$ and the ordinary generating function $h$ have many properties useful in the study of random variables, of which we can consider only a few here. of random variable X is defined as . Then, \[\begin{aligned} g(t) &=& \sum_{j = 1}^n \frac 1n e^{tj} \\ &=& \frac 1n (e^t + e^{2t} +\cdots+ e^{nt}) \\ &=& \frac {e^t (e^{nt} - 1)} {n (e^t - 1)}\ .\end{aligned}\] If we use the expression on the right-hand side of the second line above, then it is easy to see that. Improve this question. discrete uniform distribution with integer parameters a and b, where a <b. $$\int_a^b\frac{1}{b-a}e^{tx}\,dx.$$. Rectangular or Uniform distribution<br />The uniform distribution, with parameters and , has probability density function <br />. [Solved] How come i can't compute the expected value | 9to5Science Discrete Uniform Distribution Derivation of MGF (in English) . In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. \\ &=& e^{-\lambda} e^{\lambda e^t} = e^{\lambda(e^t - 1)}\ .\end{aligned}\] Then \[\begin{aligned} \mu_1 &=& g'(0) = \left. Moment Generating Function (MGF) of a Random Vector Y: The MGF of an n 1 random vector Y is given by. The moment generating function of a degenerate function is the extreme case of the uniform distribution with the upper bound infinitively approaching to the lower bound. = E( k = 0Xktk k!) What is the rationale of climate activists pouring soup on Van Gogh paintings of sunflowers? How actually can you perform the trick with the "illusion of the party distracting the dragon" like they did it in Vox Machina (animated series)? Minimum number of random moves needed to uniformly scramble a Rubik's cube? \int\limits_{-\infty}^\infty e^{tx} f( x) dx &\text{if $X$ is continuous with density $f( x)$} \sum\limits_x e^{tx} p(x) &\text{if $X$ is discrete with mass function $p( x)$}\\ Mgf, Discrete Statistical Distributions -551 (1) - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Moment generating function of a discrete distribution generating function of a discrete distribution. for a discrete distribution with N possible values: The moment generating function is written as follows: The moment generating function of the task is shown below: Then the condition of the sum of probabilities is respected: Therefore we are dealing with a random variable distribution Is there any alternative way to eliminate CO2 buildup than by breathing or even an alternative to cellular respiration that don't produce CO2? Actually, what you wrote is fairly close to what needs to be done. Section [sec 7.1]) that the range of $X$ is \[\{0,1,2,\ldots,2n\}\] and $X$ has binomial distribution \[p_Z(j) = (p_X * p_Y)(j) = {2n \choose j} p^j q^{2n - j}\ .\] Here we can easily verify this result by using generating functions. Let $p$ be the probability that the coin comes up heads, and let $q = 1-p$. }\cr}\right.\] Then the $X_k$ are independent random variables describing a Bernoulli process. We introduce the generating function $h_T(z)$ for $T$: \[h_T(z) = \sum_{n = 0}^\infty r_n z^n\ .\], Then, by using the relations above, we can verify the relation, If we solve this quadratic equation for $h_T(z)$, we get \[h_T(z) = \frac{1 \pm \sqrt{1 - 4pqz^2}}{2qz} = \frac{2pz}{1 \mp \sqrt{1 - 4pqz^2}}\ .\] Of these two solutions, we want the one that has a convergent power series in $z$ (i.e., that is finite for $z = 0$). Moment generation function (MGF) of discrete uniform distribution $M_X(t) = \dfrac{e^t (1 - e^{tN}}{N (1 - e^t)}$. 1.3.6.6.2. Uniform Distribution In terms of these moments, the mean $\mu$ and variance $\sigma^2$ of $X$ are given simply by. Mean and Variance of Discrete Uniform Distributions Show that the mean, variance, and mgf of the uniform distribution are as given in this section. $$M_X(t)=\frac{e^{10t} -1}{10(e^t-1)}.$$. Why is there a fake knife on the rack at the end of Knives Out (2019)? How to derive MGF of a uniform distribution? - Google Groups Then $g$ is uniquely determined by $p$, and conversely. Step 6 - Gives the output cumulative probabilities for discrete uniform distribution. See Answer. Creative Commons Attribution NonCommercial License 4.0. 265k 34 34 gold badges 574 574 silver badges 967 967 bronze badges. That is, M ( t) is the moment generating function (" m.g.f. $, $\begin{array}{l l} We assume, without loss of generality, that $p(x_j) > 0$ for $1 \le j \le n$, and that \[x_1 < x_2 < \ldots < x_n\ .\] We note that $g(t)$ is differentiable for all $t$, since it is a finite linear combination of exponential functions. moment generating function, population mean, variance, skewness, kurtosis. 2. The binomial distribution on $\{n,n+1,n+2,\ldots,n+k\}$. The only non-zero contribution comes from How many axis of symmetry of the cube are there? How many ways are there to solve a Rubiks cube? Stack Overflow for Teams is moving to its own domain! Legal. Uniform Distribution The mean, variance, and mgf of a - SolvedLib Denote by and their distribution functions and by and their mgfs. That is, almost all random number generators generate random . A discrete uniform distribution is one that has a finite (or countably finite) number of random variables that have an equally likely chance of occurring. Find, in terms of $g(t)$, the generating functions for, Let $X_1$, $X_2$, , $X_n$ be an independent trials process, with values in $\{0,1\}$ and mean $\mu = 1/3$. Solved Show that the mean, variance, and mgf of the | Chegg.com (probability density function) given by: P(X = x) = 1/(k+1) for all values of x = 0, . By the usual formula for expectation, In general, the density for a uniformly distributed random variable X is given by, $f(x) = 1=n , \text{ where : n is a postive integer and } x = x_1, x_2, , x_n$. Why should you not leave the inputs of unused gates floating with 74LS series logic? &= \left. Number of unique permutations of a 3x3x3 cube. Moment Generating Function for a discrete random distribution generating function of a degenerate distribution. (Discrete uniform distribution) A discrete random variable is said to be uniformly distributed. How to confirm NS records are correct for delegating subdomain? Created Date: It only takes a minute to sign up. Discrete Uniform Distribution: Variance & Mgf (malayalam). Question: Let the random variable X has the discrete uniform distribution. Be sure to note the bounds on. I don't know how to approach this with what I have from class all I can come up with is. Moment generating function for the uniform distribution. \\ &=& \frac14 + \frac12 e^t + \frac14 e^{2t}\ .\end{aligned}\] This is a polynomial in $z = e^t$, and \[h(z) = \frac14 + \frac12 z + \frac14 z^2\ .\] Hence, $X$ must have range $\{0,1,2\}$, and $p$ must have values $\{1/4,1/2,1/4\}$. \\ &=& e^{-\lambda} \sum_{j = 0}^\infty \frac {(\lambda e^t)^j}{j!} Related to the probability mass function f X(x) = IP(X = x)isanotherimportantfunction called the cumulative distribution function (CDF), F X.Itisdenedbytheformula Then: $M(t)=E(e^{tX})=\sum\limits_{x\in S} e^{tx}f(x)$. DISCRETE RANDOM VARIABLES 109 Remark5.3. Suppose now that $X$ has range $\{0,1,2,3,\ldots,n\}$ and $p_X(j) = {n \choose j} p^j q^{n - j}$ for $0 \leq j \leq n$ (binomial distribution). Now we are asked to find a mean and variance of X. 1 MOMENT GENERATING FUNCTION The m.g.f. Therefore: Building of the definition of the Moment Generating Function, $ By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Then \[\begin{aligned} g(t) &=& \sum_{j = 0}^n e^{tj} {n \choose j} p^j q^{n - j} \\ &=& \sum_{j = 0}^n {n \choose j} (pe^t)^j q^{n - j} \\ &=& (pe^t + q)^n\ .\end{aligned}\] Note that \[\begin{aligned} \mu_1 = g'(0) &=& \left. distribution, Moment generating function of a discrete Discrete Uniform Distribution Derivation of MGF (in English) I expect you to know how to compute the moment generating function of some basic random variables, like those with Bernoulli . Use MathJax to format equations. moment generating function of a distribution with multiple discrete values, The 1 11 : 06. Odit molestiae mollitia If we delete the hypothesis that $X$ have finite range in the above theorem, then the conclusion is no longer necessarily true. Your random variable $X$ takes on the values $x_1,x_2,\dots,x_n$, each with probability $\dfrac{1}{n}$. It follows from all this that if we know $g(t)$, then we know $h(z)$, and if we know $h(z)$, then we can find the $p(j)$ by Taylors formula: \[\begin{aligned} p(j) &=& \mbox{coefficient~of}\,\, z^j \,\, \mbox{in}\,\, h(z) \\ &=& \frac{h^{(j)}(0)}{j! Given a uniform distribution on [0, b] with unknown b, the minimum-variance unbiased estimator (UMVUE) for the maximum is given by ^ = + = + where m is the sample maximum and k is the sample size, sampling without replacement (though this distinction almost surely makes no difference for a continuous distribution).This follows for the same reasons as estimation for the discrete distribution . uniform distribution: A discrete distribution with two possible values can be Let $X$ be a discrete random variable with values in $\{0,1,2,\ldots,n\}$ and moment generating function $g(t)$. Let us compute the moment generating function for some of the distributions . There are basically two reasons for this. This raises a question: If $X$ is a random variable with range $\{x_1, x_2, \ldots\}$ of at most countable size, and distribution function $p = p_X$, and if we know its mean $\mu = E(X)$ and its variance $\sigma^2 = V(X)$, then what else do we need to know to determine $p$ completely? What to throw money at when trying to level up your biking from an older, generic bicycle? But a knowledge of the moments of $X$ determines its distribution function $p$ completely. PDF Chapter 5 Discrete Distributions - Department of Statistical Sciences The Discrete Uniform Distribution - Mathematics A-Level Revision This page titled 10.1: Generating Functions for Discrete Distributions is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Charles M. Grinstead & J. Laurie Snell (American Mathematical Society) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Now we shall see that the mean and variance do contain the available information about the density function of a random variable. Uniform distribution. moment generating function of a distribution with multiple discrete values. Was Gandalf on Middle-earth in the Second Age? Popularity: Medium (more popular than 90% of all packages) Description: Discrete uniform distribution moment-generating function (MGF). The variance of the Poisson distribution is easier to obtain in this way than directly from the definition (as was done in Exercise [sec 6.2]. Using the MGF show that E (X) = Expert Solution Want to see the full answer? Follow edited Feb 23, 2020 at 9:18. Expectation. Hence we choose \[h_T(z) = \frac{1 - \sqrt{1 - 4pqz^2}}{2qz} = \frac{2pz}{1 + \sqrt{1 - 4pqz^2}}\ .\] Now we can ask: What is the probability that Peter is in the lead? Note that although we sayX is 3.5 on the average, we must keep in mind that our X never actually equals 3.5 (in fact, it is impossible forX to equal 3.5). (This is called the divergence test and is the first thing to check when trying to determine whether an integral converges or diverges.). For instance, suppose $X$ and $Y$ are random variables, with distributions, \[p_X = \pmatrix{ 1 & 2 & 3 & 4 & 5 & 6\cr 0 & 1/4 & 1/2 & 0 & 0 & 1/4\cr},\] \[p_Y = \pmatrix{ 1 & 2 & 3 & 4 & 5 & 6\cr 1/4 & 0 & 0 & 1/2 & 1/4 & 0\cr}.\]. It doesn't matter. Discrete uniform distribution proof 1 answer below . Here is a more interesting example of the power and scope of the method of generating functions. This answer might be correct, but it must be clear that the mistake was in writing $$\int_{-\infty}^{+\infty}e^{tx}{1\over b-a}\,dx$$ while $$\int_{-\infty}^{+\infty}e^{tx}f(x)\,dx$$ is correct. MomentGeneratingFunction[dist, t] gives the moment-generating function for the distribution dist as a function of the variable t. MomentGeneratingFunction[dist, {t1, t2, .}] How can you prove that a certain file was downloaded from a certain website? Key Point The Uniform random variable X whose density function f(x)isdened by f(x)= 1 ba,a x b 0 otherwise has expectation and variance given by the formulae E(X)= b+a 2 and V(X)= (ba)212 Example The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0,25].Write down the formula for We know that $p$ determines $g$, since \[g(t) = \sum_{j = 1}^n e^{tx_j} p(x_j)\ .\] Conversely, assume that $g(t)$ is known. Uniform Distribution - SlideShare View Answer Q: If X has a binomial distribution with parameters n and p, derive the distribution of Y = n X and obtain its PGF and MGF. Find those integers $j$ for which $p_n(j) > 0$ from $h_n(z)$. Moment generating function of a degenerate distribution . Verify that the mean of $p_n$ is $n$ times the mean of $p$. If we compute $g'(t)/g(t)$, we obtain \[. By definition of the moment generating function: By derivative chain rule: Therefore: The moment generating function of a degenerate distribution . with multiple discrete values, The discrete distribution behind the moment Moment Generating Function of Discrete Uniform Distribution Theorem Let X be a discrete random variable with a discrete uniform distribution with parameter n for some n N . Variance is 12 NGF is M'(t) = m ( This problem has been solved! \frac {pe^t + pqe^{2t}}{(1 - qe^t)^3} \right|_{t = 0} = \frac {1 + q}{p^2}\ ,\end{aligned}\] $\mu = \mu_1 = 1/p$, and $\sigma^2 = \mu_2 - \mu_1^2 = q/p^2$, as computed in Example [exam 6.21]. So integrate from a to b. Asking for help, clarification, or responding to other answers. Let $m$ be the such value of $k$; then $S_m = 0$ and $S_k < 0$ for $1 \leq k < m$. Continuous Uniform distribution; 1.2. Find the first two moments, and hence the mean and variance, of $p_n$ from $h_n(z)$. Or else integrate from to , but use the correct density function. If we write $z = e^t$, and define the function $h$ by \[h(z) = \sum_{j = 0}^n z^j p(j)\ ,\] then $h(z)$ is a polynomial in $z$ containing the same information as $g(t)$, and in fact \[\begin{aligned} h(z) &=& g(\log z)\ , \\ g(t) &=& h(e^t)\ .\end{aligned}\] The function $h(z)$ is often called the for $X$. Find the ordinary generating functions $h_X(z)$ and $h_Y(z)$ for these distributions. So here we have 99 -0-plus 1 squared minus one, all over 12, And this comes out to 833 0.25. E(X4). Find the ordinary generating function $h_Z(z)$ for the distribution $Z = X + Y$. For this reason, it is important as a reference distribution. npm package '@stdlib/stats-base-dists-discrete-uniform-mgf' To see how this comes about, we introduce a new variable $t$, and define a function $g(t)$ as follows: \[\begin{aligned} g(t) &=& E(e^{tX}) \\ &=& \sum_{k = 0}^\infty \frac {\mu_k t^k}{k!} Show that this is always true for any probability distribution on $\{0,1,2\}$. The question is If a random variable x has the discrete uniform distribution f(x;k) = 1/k, x = 1,2,3,.k and 0 elsewhere, can it be shown that the. So the mean is given by yeah, this formula which is B plus A, over to where B is 99 A is zero, And this gives us a mean of 49.5. Discrete Uniform Distribution Class DiscreteUniform distr6 Uniform Distribution - Example and Theoretical Meaning - VEDANTU Assignment problem with mutually exclusive constraints has an integral polyhedron? Does subclassing int to forbid negative integers break Liskov Substitution Principle? To begin with, it is easy to give examples of different distribution functions which have the same mean and the same variance. I am grasping so little of this so any assistance in what a moment generating function is and the concepts needed for this question would be greatly appreciated. Step 2 - Enter the maximum value b. From the definition of the continuous uniform distribution, X has probability density function : fX(x) = { 1 b a a x b 0 otherwise. For the automatic number to work, you need to Let $X$ be a discrete random variable with finite range $\{x_1,x_2,\ldots,\linebreak x_n\}$, distribution function $p$, and moment generating function $g$. 1, Moment Moment Generating Function of Discrete Uniform Distribution Moment-Generating Function Formula & Properties - Study.com Can plants use Light from Aurora Borealis to Photosynthesize? . Generate Moments of Continuous Uniform Distribution with Moment where E( ) denotes expectation . If we consider the generation of a single random digits, then $Y$ , the number generated, is Discrete Uniform Distribution - Derivation of Mean, Variance, and MGF (Simple Version) Computation Empire. The density is 1 b a on [ a, b] and zero elsewhere. Expectation and Variance of Uniform distribution - Peace Find the moment generating function for the discrete uniform random variable X. \[p = \pmatrix{ 0 & 1 & 2 \cr 0 & 1/3 & 2/3 \cr}\ ,\] and let $p_n = p * p * \cdots * p$ be the $n$-fold convolution of $p$ with itself. This probability is given by (see Exercise $\PageIndex{10}$). Moment generating function of a What is the probability of genetic reincarnation? The moment generating function is the extreme case of a uniform distribution: Therefore: Moment generating function of a discrete distribution with two possible values . This is why `t - < 0` is an important condition to meet, because otherwise the integral won't converge. Lorem ipsum dolor sit amet, consectetur adipisicing elit. Formula The value of the expected outcomes is normally equal to the mean value for a and b, which are the minimum and maximum value parameters, respectively. 3, The Find the ordinary generating functions $h(z)$ and $h_2(z)$ for $p$ and $p_2$, and verify that $h_2(z) = (h(z))^2$. The moment generating function is defined by mgf_X(t) = E_X[exp(xt)] About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . PDF 1. The Uniform Distribution - Imperial College London Add exercises text here. Suppose $X$ has range $\{1,2,3,\ldots,n\}$ and $p_X(j) = 1/n$ for $1 \leq j \leq n$ (uniform distribution). From to a, for example, you are integrating ( 0) e t x. In other words, the rth derivative of the mgf evaluated at t = 0 gives the value of the rth moment. Restricting the set to the set of positive integers 1, 2, ., , the probability distribution function and cumulative distributions function for this discrete uniform distribution are therefore. How does DNS work when it comes to addresses after slash? But for the generating functions we have instead the simple relations \[\begin{aligned} g_Z(t) &=& g_X(t) g_Y(t)\ , \\ h_Z(z) &=& h_X(z) h_Y(z)\ ,\end{aligned}\] that is, $g_Z$ is simply the of $g_X$ and $g_Y$, and similarly for $h_Z$. Uniform distribution | Properties, proofs, exercises - Statlect Then, we take derivatives of this MGF and evaluate those derivatives at 0 to obtain the moments of x. 0 & otherwise Rubik's Cube Stage 6 -- show bottom two layers are preserved by $ R^{-1}FR^{-1}BBRF^{-1}R^{-1}BBRRU^{-1} $. This video shows how to derive the Moment Generating Function (MGF) for Discrete Uniform Distribution in English.For simple version of Discrete . Moment Generating Functions. The most important property of the mgf is the following. If $X$ and $Y$ are independent discrete random variables with range $\{0,1,2,\ldots,n\}$ and binomial distribution \[p_X(j) = p_Y(j) = {n \choose j} p^j q^{n - j}\ ,\] and if $Z = X + Y$, then we know (cf. PDF The Moment Generating Function (MGF) - Stanford University If X has the discrete uniform distribution x = 1, 2, . Show that $p$ and $p'$ have the same first and second moments, but not the same third and fourth moments. Proposition Let and be two random variables. generating function of a uniform distribution, Moment The mean will be : Mean of the Uniform Distribution= (a+b) / 2 Show that $h_Z(z)$ cannot ever have the form \[h_Z(z) = \frac{z^2 + z^3 +\cdots+ z^{12}}{11}\ .\]. Discrete Uniform Distribution Calculator - VrcAcademy Comments. PDF Lecture note on moment generating functions - Duke University Discrete Uniform Distributions - Milefoot Attempting to calculate the moment generating function for the uniform distrobution I run into ah non-convergent integral. discrete distribution behind the moment generating function of this task. Uniform Distribution - Meaning, Variance, Formula, Examples When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. In that case, This is rather convenient since all we need is the functional form for the distribution of x. The uniform distribution is generally used if you want your desired results to range between the two numbers. 1.1. MGF, Discrete Statistical Distributions - 551 | PDF - Scribd This problem has been solved! Find the ordinary and moment generating functions for $p$ and $p'$. The same is true from $b$ to $\infty$. Discrete Uniform Distribution - Derivation of Mean, Variance, and MGF Glen_b. Let its support be a closed interval of real numbers: We say that has a uniform distribution on the interval if and only if its probability density function is. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. Discrete Uniform Distribution Calculator with Examples Are witnesses allowed to give private testimonies? This page covers The Discrete uniform distribution. So integrate from $a$ to $b$. The uniform distribution is used in representing the random variable with the constant likelihood of being in a small interval between the min and the max. Mathematical and statistical functions for the Discrete Uniform distribution, which is commonly used as a discrete variant of the more popular Uniform distribution, used to model events with an equal probability of occurring (e.g. Mean is E(X) . Discrete Uniform Distribution -- from Wolfram MathWorld Or else integrate from $-\infty$ to $\infty$, but use the correct density function. Connect and share knowledge within a single location that is structured and easy to search. Lesson 20: Distributions of Two Continuous Random Variables, 20.2 - Conditional Distributions for Continuous Random Variables, Lesson 21: Bivariate Normal Distributions, 21.1 - Conditional Distribution of Y Given X, Section 5: Distributions of Functions of Random Variables, Lesson 22: Functions of One Random Variable, Lesson 23: Transformations of Two Random Variables, Lesson 24: Several Independent Random Variables, 24.2 - Expectations of Functions of Independent Random Variables, 24.3 - Mean and Variance of Linear Combinations, Lesson 25: The Moment-Generating Function Technique, 25.3 - Sums of Chi-Square Random Variables, Lesson 26: Random Functions Associated with Normal Distributions, 26.1 - Sums of Independent Normal Random Variables, 26.2 - Sampling Distribution of Sample Mean, 26.3 - Sampling Distribution of Sample Variance, Lesson 28: Approximations for Discrete Distributions, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident.