Random Variable

Random Variable

A prescription for the probability model of an experiment contains two basic ingredients: the sample space and the assignment of probability to each elementary outcome. We encountered several examples where the elementary outcomes had only qualitative descriptions rather than numerical values. For instance, with two tosses of a coin, the outcomes HH, HT, TH, and TT are pairs of letters that identify the occurrences of heads or tails. If a new vaccine is studied for the possible side effects of nausea, the response of each subject may be severe, moderate, or no feeling of nausea. These are qualitative outcomes rather than measurements on a numerical scale. Often, the outcomes of an experiment are numerical values: for example, the daily number of burglaries in a city, the hourly wages of students on summer jobs and scores on a college placement examination. Even in the former situation where the elementary outcomes are only qualitatively described, interest frequently centers on some related numerical aspects.

If a new vaccine is tested on 100 individuals, the information relevant for an evaluation of the vaccine may be the numbers of responses in the categories—severe, moderate, or no nausea. The detailed record of 100 responses can be dispensed with once we have extracted this summary. Likewise, for an opinion poll conducted on 500 residents to determine support for a proposed city ordinance, the information of particular interest is how many residents are in favor of the ordinance, and how many are opposed. In these examples, the individual observations are not numerical, yet a numerical summary of a collection of observations forms the natural basis for drawing inferences.

Focusing our attention on the numerical features of the outcomes, we introduce the idea of a random variable. Corresponding to every elementary outcome of an experiment, a random variable assumes a numerical value, determined from some characteristic pertaining to the outcome. (In mathematical language, we say that a random variable X is a real-valued function defined on a sample space.) The word “random” serves as a reminder of the fact that, beforehand, we do not know the outcome of an experiment or its associated value of X.

A random variable X associates a numerical value with each outcome of an experiment. Random variable is said to be discrete if it has either a finite number of values or infinitely many values that can be arranged in a sequence. All the preceding examples are of this type. On the other hand, if a random variable represents some measurement on a continuous scale and is therefore capable of assuming all values in an interval, it is called a continuous random variable. Of course, any measuring device has a limited accuracy and, therefore, a continuous scale must be interpreted as an abstraction. Some examples of continuous random variables are the height of an adult male, the daily milk yield of a Holstein, and the survival time of a patient following a heart attack.

For assistance with your Statistics Assignment Help you can visit classof1.com

Classof1.com is open 24/7. You can call us at 1-877-252-7763 or drop an email to learning@classof1.com