Final answer:
A random variable X is a quantitative variable based on the outcomes of a random phenomenon, and it varies according to chance. The values X can take on must be listed, and the distribution of X represents the probabilities for each possible value. Calculating probabilities and descriptive statistics like mean and standard deviation involve using the distribution.
Step-by-step explanation:
Understanding Random Variables and Their Probability Distributions
A random variable is a quantitative variable that takes on different values based on the outcomes of a random phenomenon. In the context of your question, the random variable X represents a particular quantity that we are interested in, which could range from the number of questions posted on a listserv to the number of children a person has.
Defining the Random Variable X
In words, define the random variable X: The random variable X refers to the specific measurable outcome that varies according to chance in the given situation.
List of Possible Values for X
List the values that X may take on: This step requires us to enumerate all the potential outcomes that X can assume, which will depend on the context of the problem.
Distribution of the Random Variable X
Give the distribution of X: The distribution of X provides the probabilities associated with each possible value of the random variable. To represent this, we often create a probability distribution function (PDF) or a graph.
The probability that an event occurs, such as finding the probability that no children are present or the number of people you must ask to find someone with a certain disease, is determined by this distribution.
Calculating Probabilities and Descriptive Statistics
When asked to find probabilities or descriptive statistics like the mean or standard deviation, we use the provided probability distribution. For discrete random variables, sums and summations of probabilities and their values are typically involved.