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Now we will define, discuss, and apply random variables. This will utilize and expand upon what between we haveprobability already and learned inferential about statistics. probability and will be the foundation of the bridge
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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 large population. from a largeEach ofpopulation these could or evenbe considered picking a sample one "trial" of size of a (^100) random individuals experiment. from a
The experiment. random variable must be a numeric measure resulting from the outcome of a random
If we toss a coin, the random variable might be X = the number of heads. If the we person pick one in pounds person orfrom Y = the the population, number of emergencythe random room variable visits might in the be past X = theyear. weight of
If we pick 100 individuals from a large population, the random variable might be X = the number bar = the of average diabetics weight in our of sample, individuals pโhat in =our the sample, PROPORTION or yโbar of = diabetics the average in our number sample, of xโ 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 these known processes but, under will be certain important assumptions, to understanding their processes the way can statistical be studied. inference Understanding works.
We have discussed discrete and continuous before as it related to quantitative variables. The of a definitionscontinuous are random the same variable and andin fact a discrete a continuous quantitative quantitative variable variable is a special is a special case of case a discrete random variable but we will see that the definition of a random variable is a much broader concept, especially to statisticians. In PROBABILITY this section DISTRIBUTION we will be learning for discrete about howand continuousto mathematically random define variables the andtheoretical to use this knowledge to calculate probabilities and determine what values are common or rare. We will begin with a discussion of discrete probability distributions in general and then talk more about the binomial distribution. Also, we will discuss how to calculate some values by hand and instruct you in using some tables using a (via number the course of online material calculators examples). or other For programsthe most part,such calculationsas EXCEL. We can won't be completed expect you process, to do stop any andridiculous ask us byhow hand you calculations, should be approaching so if you feel the you solution are going and through we will suchpoint a you back in the right direction! I am not a fan of meaningless labor in calculation. I am most interested in your understanding and comprehension of the concepts.
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 ears for which VARIABLE a hearing by considering aid is used. the random variable X to be equal to the number of
This = 1 if will a hearing give X =aid 0 ifis no used hearing in only aid one is used, ear (either X = 2 ifleft a hearing or right). aid is used in both ears, and X
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.
The other type of random variable is a continuous random variable. These definitions are the continuous same as variables. we had for Similarly, quantitative a continuous variables randomin data wherevariable we is hada random discrete variable and 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 more precisely. values here.So if we We consider are only Xrestricted to be the by weight our ability of a randomly or interest selected in measuring newborn. the valueWe 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 probability need to notation be able toof workthe random back and variable, forth between X. the verbal description and the
The the valuesdifference individually here is that, 2500, for 2501, continuous 2502, etc. random In fact variables, we can stillwe (^) havearen't fractions going to ofbe a listing gram which makes it impossible to list all values precisely.
Be careful, many continuous random variables may be presented as rounded values which can there lead are you TRUE to (^) gapsconclude between they theare possiblediscrete whenvalues in you fact can they observe. are continuous. If there are Ask true yourself gaps if then it is discrete. IF there are no gaps between possible values, it is continuous. One issue with these definitions is that, although the definitions work to classify random variables differently into than two its types, type. Forthere example: are some situations where we will handle a random variable
We values can 130, have 131, a rounded 132, 133, continuous and 134. randomIf those variable are my onlywhere values, we only in some have analyses,a few possible I may treat this as if it is discrete.
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.
The skills we will learn in this section on random variables will be important on our journey toward understanding statistical inference.
