6.1: Two types of Random Variables Flashcards
What is a random variable, and how is it defined?
A random variable is a numerical variable whose value is determined by the outcome of an experiment.
It assigns one and only one numerical value to each experimental outcome, representing an uncertain numerical outcome before the experiment is carried out.
How can a random variable be illustrated using an example from Sound City’s business?
Consider the experiment of selling the TrueSound-XL radio at Sound City store during a specific week. Let x represent the number of radios sold during the week.
x is a random variable because, before the week, the number of radios sold is uncertain, making it a numerical outcome that is not predetermined.
What is the distinction between a discrete random variable and a continuous random variable?
A discrete random variable is a type of random variable where the possible values can be counted or listed.
In the example given, the number of TrueSound-XL radios sold (x) could be 0, 1, 2, 3, and so on.
It can assume a finite number of values or an infinite countable sequence, such as 0, 1, 2, 3, 4, and so forth.
Provide an example of a discrete random variable based on the given information.
The number of TrueSound-XL radios sold in a week (x) at Sound City is an example of a discrete random variable because the possible values (0, 1, 2, 3, and so on) can be counted or listed, representing the different outcomes of the experiment.
Provide examples of discrete random variables mentioned in the text, and explain why they are considered discrete.
Example 1: The number x of the next three customers entering a store who will make a purchase is a discrete random variable. It can take values 0, 1, 2, or 3, representing the countable outcomes of the experiment.
Example 2: The number x of four patients taking a new antibiotic who experience gastrointestinal distress as a side effect is a discrete random variable. It can be 0, 1, 2, 3, or 4, indicating the finite and countable possibilities in this scenario.
Example 3: The number x of television sets in a sample of 8 five-year-old television sets that have not needed a single repair is a discrete random variable. It could be any value from 0 to 8, demonstrating a finite and countable range of outcomes.
Explain the concept of countable and finite values in the context of discrete random variables.
Countable and finite values mean that the possible outcomes of a random variable can be listed or counted.
In the examples given, the number of customers making a purchase, patients experiencing side effects, and functional television sets are all countable and finite because the outcomes can be explicitly listed and do not go on infinitely.
Provide examples of discrete random variables with countable and infinite values, and explain why they are considered discrete despite having infinite possibilities.
Example 4: The number x of major fires in a large city in the next two months is a discrete random variable.
It can be 0, 1, 2, 3, and so on, with no definite maximum.
Despite the infinite possibilities, it is still discrete because the outcomes can be counted and listed.
Example 5: The number x of dirt specks in a one-square-yard sheet of plastic wrap is a discrete random variable.
It can be 0, 1, 2, 3, and so forth, with no limit to the number of dirt specks. Although the possibilities are infinite, they are still countable, making it a discrete random variable.
What distinguishes a continuous random variable from a discrete random variable?
A continuous random variable can assume any numerical value within one or more intervals on the real number line.
Unlike discrete random variables, which have countable and distinct outcomes, continuous random variables have an infinite number of possible values within a given range or interval, making them continuous and uncountable.
