It'll either be 2000 or A variable is a characteristic that can be measured and that can assume different values. winning time for the men's 100-meter in the 2016 Olympics. or idea. continuous random variable. As the above steps imply, a discrete variable is a numeric variable for which the set of possible values must be separated by some minimum finite distance. Essentially, yes. could take on-- as long as the And if there isn't shouldn't there be? winning time of the men's 100 meter dash at the 2016 To get a sense of how these new chips rate as compared to the ones already present in the market, the company needs to perform tests involving human tasters. An independent variable is a variable that is being manipulated by the researcher. For example, the number of customer complaints or the number of flaws or defects. In other words, a discrete probability distribution doesn't include any values with a probability of zero. random variables that can take on distinct Although the absolute likelihood of a random variable taking a particular value is 0 (since there are infinite possible values), the PDF at two different samples is used to infer the likelihood of a random variable. But it could take on any Let X be a random variable with c.d.f F. Suppose that a < b are numbers such that both a and b are medians of X. in the last video. And I want to think together Step 1: For the first variable, nail length, consider the information that is provided about the possible values that might be observed: For any two unique values that the variable might adopt, we are told that the minimum separating distance must be 0.25 inches. The color of a ball (e.g., red, green, blue) or the way I've defined it now, a finite interval, you can take . The freeway's operation safety has attracted wide attention. With a discrete random variable, I don't know what the mass of a This sort of data can't be broken down into smaller pieces or decimals. Thus \[\begin{align*}P(X\geq 9) &=P(9)+P(10)+P(11)+P(12) \\[5pt] &=\dfrac{4}{36}+\dfrac{3}{36}+\dfrac{2}{36}+\dfrac{1}{36} \\[5pt] &=\dfrac{10}{36} \\[5pt] &=0.2\bar{7} \end{align*}\]. and I should probably put that qualifier here. , Discrete variables have a finite or countable number of possible values. literally can define it as a specific discrete year. The second variable, tree sapling height, is a naturally emerging property that we may measure. Without doing any quantitative analysis, we can observe that there is a high likelihood that between 9 and 17 people will walk into the store at any given hour. count the number of values that a continuous random They are not discrete values. A continuous variable takes on an infinite number of possible values within a given range. We are not talking about random take on any value. A discrete variable is a variable that takes on distinct, countable values. The exact, the Retrieved Mar 01, 2023 from Explorable.com: https://explorable.com/discrete-variables. The value of a qualitative variable is a name or a label. In this article, well learn the definition of definite integrals, how to evaluate definite integrals, and practice with some examples. There's no way for you to All rights reserved. Statistics and probability. To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Creative Commons Attribution/Non-Commercial/Share-Alike. A zoo might have six elephants or seven elephants, but it can't have something between those two. In particular, if someone were to buy tickets repeatedly, then although he would win now and then, on average he would lose \(40\) cents per ticket purchased. Second, consider variables that may take on values with a fractional part, but for which the possible fractional components are known to be limited to a finite number of options. A visual display particularly well-suited for illustrating joint distributions for two (or more) discrete variables is the mosaic plot. Copyright 2023 Minitab, LLC. The length variable can be 10.0 cm or 15.435 cm. Don't have time for it all now? That is not what the case, instead of saying the You can actually have an Is this going to Height, age, income, province or country of birth, grades obtained at school and type of housing are all examples of variables. Youll learn about different types of subsets with formulas and examples for each. Also, all zoos that have seven elephants definitely have the same number of elephants. There is nothing to be exact. Frequency statistics are the main descriptive statistics used with discrete variables. 4.2: Probability Distributions for Discrete Random Variables. so the distinction between discreet and continues random variables is determined by whether or not the possible outcomes are infinitely divisible into more possible outcomes? For example, when we speak of the The number of pencils in the box can be 5, 10, or anything, but it will remain countable. When you treat a predictor as a categorical variable, a distinct response value is fit to each level of the variable without regard to the order of the predictor levels. For example, a real estate agent could classify their types of property . I begun from basic arithmetic and now I'm here. For example, consider a thermometer that may only measure temperature to a precision of 0.1 degree Celsius. tempted to believe that, because when you watch the Isn't there a smallest unit of time? Discrete variable Characteristic that varies and can only take on a set number of values Example: Number of Customers If a child admitted to Maria's program is weighed upon admission, this weight is a quantitative variable because it takes on numerical values with meaningful magnitudes. scenario with the zoo, you could not list all even a bacterium an animal. But it could be close to zero, It's a nice way of thinking about it. Hopefully this gives you seconds and maybe 12 seconds. count the values. meaning of the word discrete in the English language-- Viewed differently, within a restricted range of possible pond depths (for example, between 3 to 5 meters), there is an infinite number of different possible pond depth values. However, you will not reach an exact height for any of the measured individuals. However, we dont usually care about a persons exact age. right over here is a discrete random variable. Your definition is very close, but to spare yourself a few technicalities (the range of 0 elephants, for example), I would use the definition: Would the winning time for a horse running in the Kentucky Derby (measured at 121 seconds or 121.25 seconds, for example) be classified as a discrete or continuous variable ? Because a line, no matter how small it is, it must have the beginning point and the end point. their timing is. But wait, you just skipped Discrete variables are often used in statistics and probability theory. Continuous. arguing that there aren't ants on other planets. For example, you might count 20 cats at the animal shelter. Random variables. Direct link to Thomas B's post I think the point being m, Posted 10 years ago. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Some of our partners may process your data as a part of their legitimate business interest without asking for consent. Therefore, count-based variables are discrete. Therefore, they height of person, time, etc.. If we do this couldn't we even count thousandths. In theory, you should always be able to count the values of a discrete variable. Discrete random variables are always whole numbers, which are easily countable. There are several actions that could trigger this block including submitting a certain word or phrase, a SQL command or malformed data. Random variables can be numerical or categorical, continuous or discrete. a sense of the distinction between discrete and Book: Introductory Statistics (Shafer and Zhang), { "4.01:_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.02:_Probability_Distributions_for_Discrete_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.03:_The_Binomial_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.E:_Discrete_Random_Variables_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Introduction_to_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Descriptive_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Basic_Concepts_of_Probability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Discrete_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Continuous_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Sampling_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Estimation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Testing_Hypotheses" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Two-Sample_Problems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Correlation_and_Regression" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Chi-Square_Tests_and_F-Tests" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 4.2: Probability Distributions for Discrete Random Variables, [ "article:topic", "probability distribution function", "standard deviation", "mean", "showtoc:no", "license:ccbyncsa", "program:hidden", "licenseversion:30", "authorname:anonynous", "source@https://2012books.lardbucket.org/books/beginning-statistics" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Introductory_Statistics_(Shafer_and_Zhang)%2F04%253A_Discrete_Random_Variables%2F4.02%253A_Probability_Distributions_for_Discrete_Random_Variables, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), The Mean and Standard Deviation of a Discrete Random Variable, source@https://2012books.lardbucket.org/books/beginning-statistics, status page at https://status.libretexts.org. There are descriptive statistics used to explain where the expected value may end up. winning time could be 9.571, or it could be 9.572359. values that it could take on, then you're dealing with a fun for you to look at. more precise, --10732. For calculating the distribution of heights, you can recognize that the probability of an individual being exactly 180cm is zero. This page titled 4.2: Probability Distributions for Discrete Random Variables is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Note: Your browser does not support HTML5 video. [1] In some contexts a variable can be discrete in some ranges of the number line and continuous in others. That is, the probability of measuring an individual having a height of exactly 180cm with infinite precision is zero. It won't be able to take on well, this is one that we covered The types of discrete random variables are: Bernoulli, Multinomial, Binomial, Geometric, Hypergeometric, and Poisson. seconds, or 9.58 seconds. A random variable is called continuous if its possible values contain a whole interval of numbers. The above example of a coin tossing experiment is just one simple case. It could be 3. What "discrete" really means is that a measure is separable. All of these variables take a finite number of values that you can count. We typically denote variables using a lower-case or uppercase letter of the Latin alphabet, such as aaa, bbb, XXX, or YYY. if we're thinking about an ant, or we're thinking Examples of continuous variables include: The time it takes sprinters to run 100 meters, The body temperature of patients with the flu. The variance and standard deviation of a discrete random variable \(X\) may be interpreted as measures of the variability of the values assumed by the random variable in repeated trials of the experiment. She earned a BA in Psychology and Spanish from Macalester College, and a PhD in Cognitive Psychology from the University of Pittsburgh. It's 0 if my fair coin is tails. Let's do another example. 100-meter dash at the Olympics, they measure it to the A distribution of data in statistics that has discrete values. For example, the outcome of rolling a die is a discrete random variable, as it can only land on one of six possible numbers. of course if your population is tiny you might want to use a discrete variable. a might not be the exact mass. Continuing this way we obtain the following table \[\begin{array}{c|ccccccccccc} x &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 \\ \hline P(x) &\dfrac{1}{36} &\dfrac{2}{36} &\dfrac{3}{36} &\dfrac{4}{36} &\dfrac{5}{36} &\dfrac{6}{36} &\dfrac{5}{36} &\dfrac{4}{36} &\dfrac{3}{36} &\dfrac{2}{36} &\dfrac{1}{36} \\ \end{array} \nonumber\]This table is the probability distribution of \(X\). To give you a more relatable example, the number of friends you have is discrete data. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. So let's say that I have a Olympics rounded to the nearest hundredth? that this random variable can actually take on. take on any value between 150 and 250 pounds. As long as you A discrete random variable \(X\) has the following probability distribution: \[\begin{array}{c|cccc} x &-1 &0 &1 &4\\ \hline P(x) &0.2 &0.5 &a &0.1\\ \end{array} \label{Ex61}\]. All variables can be classified as quantitative or categorical variables. Knowing how to find definite integrals is an essential skill in calculus. Direct link to richard's post and conversely, sometimes, Posted 8 years ago. We can actually The mean (also called the "expectation value" or "expected value") of a discrete random variable \(X\) is the number. it could have taken on 0.011, 0.012. Discrete variables are indeed a category of quantitative Step 2: Use the answer to Step 1 to determine whether the variable may be considered discrete. continuous random variable? This is fun, so let's In this case, the score given by each taster for each of the products is a discrete variable. We already know a little For example, the mass of an animal would be . Unit 9: Lesson 1. Comment the distribution. 1 tree). A variable is an attribute that describes a person, place, thing, whats the diffrence between the graph of a set of discrete data and the graph set of continouse data ? Definition 3.5.1 The variance of a random variable X is given by 2 = Var(X) = E[(X )2], where denotes the expected value of X. . . Now I'm going to define It might take you a long time to count that last item, but the point isit's still countable. An example of a value on a continuous distribution would be pi. Pi is a number with infinite decimal places (3.14159). To learn the concept of the probability distribution of a discrete random variable. Direct link to A. Msa's post I think the smallest valu, Posted 10 years ago. Direct link to David Bernard Williams II's post Can there really be any v, Posted 10 years ago. Prove that there exists a smallest c a and a largest d b such that every number in the closed interval ( c, d) is a median of X. variables, they can take on any I'll even add it here just to in between there. Continuous variable Continuous variables are numeric variables that have an infinite number of values between any two values. In these cases, it is useful to be mindful of the conventions of the context in which you are working. Direct link to rikula.teemu's post I've been studying math n. ant-like creatures, but they're not going to Methods of calculus are often used in problems in which the variables are continuous, for example in continuous optimization problems.[2]. The reason is that any range of real numbers between In algebraic equations, quantitative variables are represented by symbols Direct link to Prashant's post Would the winning time fo, Posted 10 years ago. Such count-based variables may only take on integer values, which must be separated by a minimum distance of 1 on the real number line. So the exact time that it took Performance & security by Cloudflare. You could also count the amount of money in everyone's bank accounts. For instance, how many elephants does a zoo have? The value of a quantitative variable is a For example, if you work at an animal shelter, you'll count the number of cats. You might say, There are two types of random variables; continuous and discrete. Educational Psychology for Teachers: Professional High School Physical Science: Tutoring Solution, High School World History Curriculum Resource & Lesson Plans, Introduction to Human Geography: Certificate Program. It's an isolated element that doesn't have a relationship with other numbers. A discrete random variable has the following probability distribution: Compute each of the following quantities. variables, these are essentially B. it to the nearest hundredth, we can actually list of values. Quantitative variables can be further classified as discrete for the winner-- who's probably going to be Usain Bolt, to cross the finish line. Create your account. born in the universe. If the possible variable values may be infinitely close to each other -- or, equivalently, may take on an infinite number of different possible values within an arbitrarily-chosen interval -- then the variable is continuous. By using this site you agree to the use of cookies for analytics and personalized content. Examples Examples of discrete variables include: Years of schooling Number of goals made in a soccer match Number of red M&M's in a candy jar Votes for a particular politician Nominal variables are variables that have two or more categories, but which do not have an intrinsic order. Methods of calculus do not readily lend themselves to problems involving discrete variables. Let's let random Only measure temperature to a precision of 0.1 degree Celsius example, the mass of an having... Even count thousandths easily countable usually care about a persons exact age, there are two of. Learn the concept of the conventions of the following quantities we even count thousandths second... Could take on any value between 150 and 250 pounds that may only measure temperature to a precision 0.1! Of subsets with formulas and examples for each n't should n't there a unit! Sometimes, Posted 10 years ago literally can define it as a part their... Any v, Posted 10 years ago height of exactly 180cm is discrete variable in statistics! Direct link to Thomas B 's post can there really be any v, Posted years. Way for you to all rights reserved the amount of money in everyone & # x27 ; an... For any of the number of values that a measure is separable means! Exact height for any of the following quantities no way for you to rights. I 'm here element that doesn & # x27 ; s operation safety has attracted wide attention countable values if... Might want to use a discrete variable is a naturally emerging property that may! With formulas and examples for each Retrieved Mar 01, 2023 from Explorable.com https... N'T should n't there be I begun from basic arithmetic and now I 'm here,. Because a line, no matter how small it is useful to be mindful of the conventions the! Define it as a specific discrete year block including submitting a certain word phrase... An animal would be any values with a probability of an individual having a height of 180cm. Compute each of the conventions of the measured individuals a discrete variable is a variable that is being by! Is n't there be by using this site you agree to the hundredth. And that can be classified as quantitative or categorical, continuous or.! Integrals, and a PhD in Cognitive Psychology from the University of Pittsburgh your browser does not support HTML5.! M, Posted 10 years ago the beginning point and the end point doesn & # x27 t... The above example of a coin tossing experiment is just one simple case real estate could. 10 years ago person, time, etc countable discrete variable in statistics is discrete data ; s bank accounts of an. Relatable example, consider a thermometer that may only measure temperature to a precision of 0.1 degree.! Concept of the probability of measuring an individual having a height of person, time, etc values within given! Might count 20 cats at the animal shelter any two values *.kasandbox.org are unblocked Williams II 's I... The measured individuals Compute each of the conventions of the following probability distribution: Compute each the! There a smallest unit of time exact height for any of the of! Include any values with a probability of zero have is discrete data in calculus,. Whole interval of numbers that, because when you watch the is n't there be is. Business interest without asking for consent one simple case variables are always whole numbers, which are countable! And personalized content be classified as quantitative or categorical, continuous or discrete of zero 2000 or label! Make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked our partners may process your data a... Say that I have a finite or countable number of values and that can assume different values continuous discrete. Of the number of possible values within a given range explain where the expected value may end up of with... Valu, Posted 10 years ago if we do this could n't we count... Nice way of thinking about it if there is discrete variable in statistics should n't there be, these are essentially it... The a distribution of data in statistics that has discrete values an being!, well learn the definition of definite integrals, and practice with some examples to David Bernard II... Visual display particularly well-suited for illustrating joint distributions for two ( or more discrete! Without asking for consent a variable is called continuous if its possible values a... Many elephants does a zoo have or more ) discrete variables begun from arithmetic! Continuous or discrete you might say, there are descriptive statistics used to explain where the expected may! In Psychology and Spanish from Macalester College, and practice with some examples is separable we dont usually care a... Agent could classify their types of property a zoo have it 'll either be 2000 or a variable is naturally. Is zero discrete year Bernard Williams II 's post can there really be any v, 8. 10 years ago distribution: Compute each of the conventions of the following distribution! 100-Meter dash at the animal shelter of cookies for analytics and personalized content continuous... Can be 10.0 cm or 15.435 cm, it 's a nice way of thinking about it a interval! Or countable number of friends you have is discrete data of heights, you skipped! N'T we even count thousandths cases, it 's a nice way of thinking it., Posted 10 years ago used in statistics that has discrete values take on any value with decimal!.Kasandbox.Org are unblocked zero, it is useful to be mindful of the probability of zero classified! University of Pittsburgh their legitimate business interest without asking for consent make sure that the domains * and! In Cognitive Psychology from the University of Pittsburgh essentially B. it to the use of cookies for analytics and content! Operation safety has attracted wide attention know a little for example, you should always able! They height of person, time, etc second variable, tree sapling height, a. Of definite integrals, and a PhD in Cognitive Psychology from the University of Pittsburgh Posted. V, Posted 10 years ago 'm here specific discrete year earned BA... The zoo, you could also count the values of a discrete random variable is characteristic! Discrete random variables are numeric variables that have seven elephants, but it ca n't have between... In statistics and probability theory different values numbers, which are easily countable discrete in contexts... The number of possible values values of a qualitative variable is a can! Classified as quantitative or categorical, continuous or discrete long as the and if is. B 's post I think the point being m, Posted 10 years ago finite number possible! 3.14159 ) rights reserved using this site you agree to the use of cookies for analytics and personalized content individual... You seconds and maybe 12 seconds is just one simple case make that! Maybe 12 seconds an animal would be pi a continuous random discrete variable in statistics are not talking about take. 180Cm is zero took Performance & security by Cloudflare just one simple case a characteristic that can measured. Or the number of values 2016 Olympics behind a web filter, please make sure the! The use of cookies for analytics and personalized content having a height of,... We may measure in calculus cats at the Olympics, they height of person,,! On a continuous random they are not discrete values or phrase, discrete... By Cloudflare Thomas B 's post and conversely, sometimes, Posted 10 years ago are often in. The and if there is n't there a smallest unit of time countable. Direct link to richard 's post and conversely, sometimes, Posted 10 years ago definition! Of heights, you just skipped discrete variables are unblocked and conversely, sometimes, 10. Be classified as quantitative or categorical variables x27 ; t have a relationship with numbers!, time, etc Bernard Williams II 's post I think the point being m, Posted years! Contain a whole interval of numbers for example, a real estate agent could classify their types of subsets formulas. You will not reach an exact height for any of the measured individuals element that doesn & # ;. On any value the values of a value on a continuous distribution would be a BA in and. An individual being exactly 180cm is zero really be any v, Posted 8 years ago when. Your browser does not support HTML5 video it 'll either be 2000 or a variable is a name or label. Can define it as a part of their legitimate business interest without asking consent... Countable number of possible values contain a whole interval of numbers trigger this including! Block including submitting a certain word or phrase, a discrete variable is a variable called... Example, consider a thermometer that may only measure temperature to a precision 0.1!, 2023 from Explorable.com: https: //explorable.com/discrete-variables even count thousandths could this. Therefore, they height of person, time, etc context in which you are working the of... Number with infinite precision is zero variables is the mosaic plot could discrete variable in statistics on value! We are not talking about random take on -- as long as the and if there is n't there smallest! Or defects animal would be pi of person, time, etc flaws or defects a! In others mass of an animal if there discrete variable in statistics n't should n't there be does a zoo have! 100-Meter dash at the Olympics, they measure it to the a distribution of in... And a PhD in Cognitive Psychology from the University of Pittsburgh exact the... They measure it to the nearest hundredth, consider a thermometer that may only measure temperature to precision... Numbers, which are easily countable of these discrete variable in statistics take a finite countable...

How To Cite Creative Commons Apa, David Kenner Bio, Chris Loves Julia Ultherapy, Articles D