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Understanding Random Variables: Discrete and Continuous, Lecture notes of Statistics

An introduction to random variables, their types, and their applications in statistics. It covers discrete and continuous random variables, their definitions, examples, and probability distributions. The document also discusses how to convert categorical variables into random variables and how to determine probabilities for discrete and continuous random variables. Students will learn the importance of understanding random variables in the context of statistical inference.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

eklavya
eklavya 🇺🇸

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Download Understanding Random Variables: Discrete and Continuous and more Lecture notes Statistics in PDF only on Docsity! Now we will define, discuss, and apply random variables. This will utilize and expand upon  what we have already learned about probability and will be the foundation of the bridge  between probability and inferential statistics.   We will develop the theoretical background for some simple situations and then we won't  focus too much on the theory once the mathematics becomes more difficult.   Our goal is to give you a good feel for why probability and random variables are useful on  their own and then use a few simple examples to help you understand how statistics really  works.   Finally, at the end of the semester, we will learn to apply the process of statistical inference  using numerous common statistical tests for hypotheses involving one and two variables,  such t‐tests, analysis of variance, and chi‐squared tests.  The foundations we have been building will be important to the development of p‐values  and confidence intervals which, you may already know, are the basis of our ability to draw  conclusions about our population from data.  1 We need to define what we mean by a random variable. In statistics, a random variable  assigns a unique numeric value to the outcome of a random experiment.   The term "random experiment" is very broad, anything from tossing a coin to picking an  individual from a large population or even picking a sample of size 100 individuals from a  large population. Each of these could be considered one "trial" of a random experiment.     The random variable must be a numeric measure resulting from the outcome of a random  experiment.  If we toss a coin, the random variable might be X = the number of heads.   If we pick one person from the population, the random variable might be X = the weight of  the person in pounds or Y = the number of emergency room visits in the past year.   If we pick 100 individuals from a large population, the random variable might be X = the  number of diabetics in our sample, p‐hat = the PROPORTION of diabetics in our sample, x‐ bar = the average weight of individuals in our sample, or y‐bar = the average number of  emergency room visits in the past year in our sample.   All of these measures are what statisticians would call random variables. Their values are  not known but, under certain assumptions, their processes can be studied.  Understanding  these processes will be important to understanding the way statistical inference works.  2 In a study of individuals with some degree of hearing loss, individuals were asked in which  ear(s) they wear a hearing aid. Possible answers were none, left, right, both.   As recorded, this is a categorical variable.  However, we can convert it to a numeric  RANDOM VARIABLE by considering the random variable X to be equal to the number of  ears for which a hearing aid is used.   This will give X = 0 if no hearing aid is used, X = 2 if a hearing aid is used in both ears, and X  = 1 if a hearing aid is used in only one ear (either left or right).   So this is an example of a discrete random variable.  It has three possible values with gaps  between them.  We can list the possible values and they are numeric.   5 Example - Discrete = X is a quantitative variable which takes the possible values of 0, 1, or 2. = We can ask questions like: » What is the probability that a randomly selected person will have a hearing aid in both ears? » What is the probability that a randomly selected person will not be wearing a hearing aid in either ear? » What is the probability that a randomly selected person will have a hearing aid only one ear? UF NIVERSITY of We can ask questions like: What is the probability that a randomly selected person from our sample will use hearing aids in both ears? Neither ear? Only one ear? Here we know the maximum number we could observe is 2. We can have discrete random variables where we do not know the upper (or lower) limit. We cannot place a clear maximum on the possible outcome. However, there are still gaps between the values and they can be listed. The other type of random variable is a continuous random variable.  These definitions are  the same as we had for quantitative variables in data where we had discrete and  continuous variables.  Similarly, a continuous random variable is a random variable that can  take on any value in an interval. There are no longer any gaps between the possible values.   Suppose we consider the weight of newborn infants in grams. We cannot list all of the  possible values here. We are only restricted by our ability or interest in measuring the value  more precisely. So if we consider X to be the weight of a randomly selected newborn.  We  can ask questions like  What is the probability that X will be less than 2500 grams? In other words, what is the  probability that the newborn will weigh less than 2500 grams?   What is the probability that X will be between than 2800 and 3400 grams? In other words,  what is the probability that the newborn will weigh between 2800 and 3400 grams? We need to be able to work back and forth between the verbal description and the  probability notation of the random variable, X.   The difference here is that, for continuous random variables, we aren't going to be listing  the values individually 2500, 2501, 2502, etc.  In fact we can still have fractions of a gram  which makes it impossible to list all values precisely.   7 Comments = Agood rule of thumb is that discrete random variables are things we count, while continuous random variables are things we measure. = We counted the number of ears in which a patient wear's a hearing aid. This was a discrete random variable. = We measured the weight of a newborn. This was a continuous random variable. UF NIVERSITY of A good rule of thumb is that discrete random variables are things we count or list while continuous random variables are things we measure. We counted the number of ears in which a patient wear's a hearing aid. This was a discrete random variable. We measured the weight of a newborn. This was a continuous random variable. 10 INTRODUCTION Unit 3B: Random Variables UF HiORiDs The skills we will learn in this section on random variables will be important on our journey toward understanding statistical inference. 11