A discrete random variable must have a finite range. ) can have a finite or infinite number of values.

A discrete random variable must have a finite range. infinite sequence b. ) must have a clear upper limit. Oct 6, 2018 · Note that number of telephone calls arriving at an office in a finite time is an example of discrete random variable. The probability distribution of a discrete random variable X is a list of each possible value of X together with the probability that X takes Discrete random variables have a set of possible values that are either finite or countably infinite. When the range of X is infinite there is a possibility that the infinite series will not be absolutely convergent and therefore th Random variables are essential in probability, categorized as discrete or continuous. To summarize, the difference between discrete and continuous probability distributions has to do with the nature of the random variables they represent. There is a notion of independence of a random variable from an event or from another random variable. For example, the number of times heads comes up when flipping a coin 10 times is a discrete random variable, since it can have Continuous random variables take values in an interval of real numbers, and often come from measuring something. - are countable. Types of random variables ¶ Due to the nature of random events, they can be discrete or continuous. Proof: Let X: Ω → R X: Ω → R be a random variable. 3. Study with Quizlet and memorize flashcards containing terms like a _____ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure, a _____ random variable has either a finite or a countable number of values, a ________ random variable has infinitely many values associated with measurements and more. 3 Expectation We have this idea of a random variable, which is actually neither random nor a variable (it's a deterministic function X : ! X. Random variables allow characterization of outcomes, so that we do not need to focus on each outcome specifically. Working with discrete random variables requires summation, while continuous random variables require integration. However, there exists another group of random variables that can assume an uncountable set of possible values. Aug 11, 2020 · Classify as true or false and explain why for each discrete probability scenario (a) If $X,Y$ are discrete random variables (both with a finite range) satisfying If a random variable can take only a finite number of distinct values, then it must be discrete. 4. However, in this case, it seems that the range of a continuous random variable is not necessarily uncountable. it must be possible to list all the values in a, possibly infinite, sequence). Probability distributions summarize the likelihood of outcomes, requiring that probabilities are between 0 and 1 and sum to 1. A discrete variable is defined as a random variable that can only take on a finite number of values or an infinite number of countable values. We have an expert-written solution to this problem! Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval. Discontinuities in CDFs occur atvalues x which occur with positive probability, so that continuous randomv riables, we want the pr tin Jun 13, 2025 · Applications of Discrete Random Variables Discrete random variables have numerous applications in fields such as finance, engineering, and computer science. Unlike discrete random variables, a continuous random variable must have a continuous cumulative distribution function (CDF FX(x), as illustrated in Figure 3. Notice that in each of the examples above, the space \ (R\) was finite. ) can have a finite or infinite number of values. Definition \ (\PageIndex {1}\) A discrete random variables refers to variables that has either finite or a c ountable number of possible values. To be able to apply the material learned in this lesson to new problems. A random variable is called continuous if … Probability Distributions for Discrete Random Variables Probabilities assigned to various outcomes in the sample space S, in turn, determine probabilities associated with the values of any particular random variable defined on S. A random variable is continuous if both of the following apply: Study with Quizlet and memorise flashcards containing terms like What is a discrete random variable?, What is a continuous random variable?, How is a discrete random variable modelled vs a continuous random variable? and others. , A random variable is said to be continuous if it, One condition of a well-defined probability density function of a continuous random variable X is that f(x) is and more. , a random experiment). These variables contrast with continuous random variables by assuming a finite or countably infinite set of outcomes. Study with Quizlet and memorize flashcards containing terms like Random Variable, Probability Distribution, Discrete Random Variable and more. Understanding discrete random variables is crucial for grasping the principles of probability Random variables are essential in probability distributions, categorized as discrete or continuous. Examples of discrete random variables include the number of children in a family, the friday night attendance at a cinema, the number of patients in a doctors surgery, the number of defective light bulbs in a box of ten. Dec 31, 2020 · I wrote No, how else can outcomes be measured if they aren't finite or countably infinite (which falls under discrete random variables) or if they can't be measured using a range of values (which falls under continuous random variables)? Study with Quizlet and memorize flashcards containing terms like Which of the following represents a difference between continuous and discrete random variables?, Which of the following is always true for all probability density functions of continuous random variables?, The probability density function, f(x), for any continuous random variable X, represents: and more. A discrete random variable is a random variable that can only take on a finite number or infinite range of values; a continuous number is a number that can take any value within a specific range on the line of real numbers. For example, the sample mean of samples drawn from a Cauchy distribution has the same (Cauchy) distribution as the individual samples. These values can typically be listed out and are often whole numbers. ) are countable. The probability distribution of the discrete variable is known as probability mass function. A random variable is defined as a function that associates outcomes of an experiment with real numbers. has a countable number of values. Jul 23, 2025 · Discrete Random Variable Definition In probability theory, a discrete random variable is a type of random variable that can take on a finite or countable number of distinct values. For example, X could be the random variable which represents the number of A random variable is said to be discrete if its range of values is finite or countably infinite (i. can have decimal values. It can be discrete or continuous, depending on whether the outcomes are finite or infinite, forming the basis for probability distributions and density functions. , The probability that a continuous random variable equals any of its values is, A random variable is said to be discrete if it has _____ and more. We may have either a discrete random variable, a continuous random variable or a mixed random variable. random variable. Such random vari bles have the heorem 4. - have a set of distinct values - can have a finite or infinite number of values. A discrete random variable can have a finite or infinite number of outcomes. Any discrete random will have a well-defined expected value. The main difference between discrete and continuous variables is that discrete variables can only take on specific values, whereas continuous variables can take on any value within a specified range. A random variable is called discrete if its possible values form a finite or countable set. A random variable is a number generated by a random experiment. The binomial distribution and the geometric distribution are the examples of the discrete distributions. 2. It represents whole units or categories, where fractional or partial values are not possible. To learn the formal definition of a discrete random variable. Discrete random variables result from counting distinct outcomes, while continuous random variables can take on an infinite range of measurable values. 1: Random Variables A random variable is a number generated by a random experiment. e. 1. Probability Distributions for Discrete Random Variables Probabilities assigned to various outcomes in the sampe space S, in turn, determine probabilities associated with the values of any particular random variable defined on S. For a finite (discrete) random variable, state the two requirements for p_k to be a valid probability distribution. discrete random variable d. A random variable is called continuous if its possible values contain a whole interval of numbers. A discrete random variable has a finite or countably infinite number of possible outcomes. AI generated definition based on: Computer Systems Performance Evaluation and Prediction, 2003 4. , The probability that a discrete random variable equals any of its values is, A random variable is said to be continuous if it Multiple select question. We begin with the formal definition. As some examples, consider the following: The sum of values shown when two dice are rolled The fraction of area covered by crab grass in a randomly selected lawn The number of heads seen when flipping a coin 3 times The number The range of this random variable is countable, so Y is a discrete random variable. 5. Continuous probability distributions deal with random variables that can take on any value within a given range or interval. 1. For a discrete random variable X, the probability mass function (PMF), pX (x), is defined as follows: 3. For example, the number of children in a family can be represented using a discrete random variable. These values are typically real numbers, and the range can be either bounded or unbounded. If Ω Ω is finite or countably infinite Jul 27, 2024 · It confused me badly. For example, the number of students in a classroom is a discrete variable because you cannot have a fractional student—it must be a whole number. 3. The number can be finite or infinite; that is, the random variable can have a countably finite number of values or a countably infinite number of values. You might want to know what you might expect it to equal on average. Any finite discrete random variable will have a well-defined expected value 2. To be able to use the probability mass function of a hypergeometric random variable to find probabilities. We have seen that an intuitive way to view the probability of a certain outcome is as the frequency with which that … Discrete Random Variables A discrete random variable is a variable that can take on a finite or countably infinite number of values. If Ω Ω is a finite or countably infinite sample space then every random variable X: Ω → R X: Ω → R must be a discrete random variable. By a Continuous Random Variable I think they mean the following: If the sample space $\Omega$ is a continuum, then the the random variable defined on $\Omega$ is called a continuous random variable. We would like to show you a description here but the site won’t allow us. has Jul 23, 2025 · A random variable can assign a number (like 1 to 6) to each of these outcomes, allowing us to analyze the results using statistical methods. a measure of the average, or central value of a random variable b. Therefore, each of those random variables would be considered countable. One can however have random variables with a infinite number of values in \ (R\) and yet still be countable. t for a more precisedefinition later. discrete probability function c. If that's your question (and it looks almost identical), then you already got good answers there (with examples). Any continuous random variable will have a well-defined expected value 4. - Discrete: the random variables assumes a countable number of distinct values. In Part III we will focus on the continuous world. A random variable that can assume only a finite number of values is referred to as a (n) a. Study with Quizlet and memorize flashcards containing terms like True or false: A continuous random variable can have a finite set of integer values. a measure of the dispersion of a random variable Random variables ¶ A random variable is a numerical representation of a random event. A discrete random variable is a variable that can take on a finite number of distinct values. Generally, a random variable is called continuous if its possible values contain a whole interval of numbers. Exact time of arrival of a call is an example of continuous random variable. There are two possible outcomes, modeled as random variables We define a random variable as a function that maps from the sample space of an experiment to the real numbers. Jun 13, 2018 · Therefore, a discrete random variable does not have to have a 'finite number of options', but there needs to be a non-infinitesimal gap between the possible values. Therefore, the answer is (b) False. x = 2. Find the mean and variance of a discrete random variable, and apply these concepts to solve real-world problems. (t/f), Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval. For example, the number of heads in 10 coin flips (finite) or the number of times you can flip a coin until it lands on heads (potentially infinite). Mar 26, 2023 · A random variable is a number generated by a random experiment. Its range is the set of rational numbers, every rational number has positive probability, and the set of irrational numbers has zero probability. Using geometric random variables If one has a sequence of independent experiments where the probability of “success” being p, then the random variable which expresses the probability of n failures before the first success is geometrically distributed with success probability p. Unlike discrete random variables, which have countable outcomes, continuous random variables are associated with measurable and Discrete Random Variables In Part I, we saw that experiments are classified as either having a discrete sample space, with a countable number of possible outcomes, or a continuous sample space, with an uncountable number of possible outcomes. To understand the conditions necessary for using the hypergeometric distribution. Discrete random variables are fundamental components in statistics, representing values that result from counting outcomes of an experiment, such as the number of heads in a coin toss. We will discuss these two types of random variables separately in this chapter and in Chapter 4. Discrete Random Variables: The discrete random variables are those which can take countable number of values in the range of the variable. 45734 pounds of apples. AI generated definition based on: Data Handling in Science and Technology, 2003 Aug 16, 2025 · In this section, we introduce and discuss the framework of random variables and develop the basic tools for working with discrete random variables. The range of X is X = f0; 1; : : : ; ng because there could be any where from 0 to n heads is a discrete random variable because there are nite n + 1 values that it takes on. When it comes to random variables, there are three classes of random variables. When the range of X is finite, E[X] a ways exists since it is a finite sum. These values are typically integers or whole numbers, although they can also be other types of values, such as categories or labels. Let X be the random variable that represents the number of heads in a single coin flip. To learn the formal definition of a discrete probability mass function. To find the mean of a discrete A discrete random variable is a type of random variable that can take on a countable number of distinct values. These values are often represented by integers or whole numbers, other than this they can also be represented by other discrete values. . is measured over an interval. Jul 23, 2025 · What is a Continuous Random Variable? Continuous random variable is a type of random variable that can take on an infinite number of possible values within a given range. Discrete random variable: Can take a countable, finite number of distinct values. Variance and standard deviation measure the spread of Apr 19, 2014 · Distributions with infinite variance are heavy-tailed; there are lots of outliers, and can have properties that are different from what one is used to seeing. In this section, we'll explore some of the key applications, including expected value and variance, common discrete probability distributions, and real-world examples. ) However, the way I like to think about it is: it a random quantity which we do not know the value of yet. Discrete random variables have distinct outcomes, like dice rolls, while continuous random variables can take any value within a range, such as heights. For an infinite (continuous) random variable, state the two requirements for f (x) to be a valid probability density function (PDF). 2: Probability Distributions for Discrete Random Variables The probability distribution of a discrete random variable If a random variable can take only a finite number of distinct values, then it must be discrete. A random variable is called continuous if … Sep 12, 2025 · We are now in a position to prove our first fundamental theorem of probability. discrete random variable Variance is a. Aug 5, 2019 · Also see answers to this question on reddit only about a day ago: Must a discrete random variable have a finite range?. A discrete random variable is a random variable whose possible values either constitute a finite set or else can be listed in an infinite sequence. If a random variable can take only a finite number of distinct values, then it must be discrete. Random Variables / Discrete Random Variables The idea of a random variable starts with a numerical value determined by some chance process (i. - Continuous: the random variable is characterized by (infinitely) uncountable values within any interval. For example, the random variables mentioned in (a) and (b) above can take at most a finite number of numerical values, and are therefore discrete. Must a discrete random variable have a finite range? I was wondering about real life examples of discrete random variables with infinite ranges. 2. Study with Quizlet and memorize flashcards containing terms like A continuous random variable can have a finite set of integer values. o E[c] = c · P (c = c) = c · 1 = c. well as the constant random variable. Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten. In this part, our focus will be on the discrete world. May 13, 2022 · The statement about discrete random variables is false because they can take on a finite or countable number of values, not an infinite number. random variable is called discrete if its range (the set of values that it can take) is finite or at most countably infinite. 1: Random Variables Learning Objectives Distinguish between discrete and continuous random variables Find the probability distribution of discrete random variables, and use it to find the probability of events of interest. A discrete random variable is a variable that can only take on certain exact values; they can either take on a finite number of distinct values, or a countably infinite set of values, like the integers. This data type often occurs when you are counting the number of event occurrences. Once we know how to deal with one branch of random variables, the theory concerning the other two branches are very similar. This is quite different from the usual belief that the sample mean is a better "estimator" than any Sep 8, 2025 · We have seen that the mean and variance of a random variable contain important information about the random variable, or, more precisely, about the distribution function of that variable. (T/F), A Dec 21, 2022 · The statement you give can be viewed as a theorem (if p p then q q) rather than a definition of q q: Theorem: Fix probability space (Ω,F, P) (Ω, F, P). ) have a set of distinct values. Study with Quizlet and memorize flashcards containing terms like A RANDOM VARIABLE, A DISCRETE RANDOM VARIABLE, The expected value E(X) of a discrete random variable and more. If a continuous random variable has non-zero probability only on a closed interval [a,b] then it will have a well-defined Study with Quizlet and memorize flashcards containing terms like Since there is an infinite number of values a continuous random variable can assume, the probability of each individual value is virtually 0. finite sequence c. Definition \ (\PageIndex {3}\) Now that we have formally defined probability and the underlying structure, we add another layer: random variables. To find the mean of a discrete random variable, use the equation: μ = ∑ x P (x). A random variable is continuous if it can assume an uncountable number of values, for example, any value from a certain interval. The range is a subset of all real numbers If the range assumes values from a countable set (i. Mar 25, 2024 · Discrete Variable A discrete variable is a quantitative variable that assumes a finite or countable number of values. A discrete random variable has distinct values that are countable and finite or countably infinite. We will use capital letters towards the end of the alphabet such as \ (X, Y, Z\) to represent random variables. A random variable is discrete if it can assume only a countable (or finite) number of values. , takes on only a finite number of values), the random variable is then a discrete random variable. Discrete random variables 1. If you still have remaining questions after reading the linked posts and Wikipedia's list, please post a new question. Discrete probability distributions are associated with variables that take on a finite or countably infinite number of distinct values. hzdybgw svwl hikxk xactcn bvodf jrthrat wrq rminqhr tapxkfvt ggrqf