The domain in the context of a library's book collection is the number of years, set between 0 to 50, whereas the range is the number of books, between 500 to 10,000. Random variables represent values that can change based on outcomes, and the probability distribution of these variables has two key characteristics: probabilities summing to one and each individual probability between zero and one.
Understanding Domain and Range
The concept of domain refers to all possible input values which the function may accept. For typical functions, like the one representing the growth of a library's book collection over time, this would be the number of years that have passed. In the provided scenario, we are considering the domain to be from 0 to 50 years since the library's inception.
On the other hand, range corresponds to the possible output values that a function can produce. Considering the context of library books, if the function represents the number of books in the library, and we know that the library starts with 500 books and can grow up to 10,000, the range would be from 500 to 10,000 books.
Probability Distribution Function and Variables
In probability and statistics, a random variable represents a value that can change within its domain based on outcomes of a random phenomenon. For example, the domain for a student's major (X), class count (Y), and money spent (Z) varies based on the individuals and their activities. These are considered discrete random variables because they can take on a specific set of values.
Furthermore, the discrete probability distribution quantifies the probability of each possible value of a discrete random variable. Two essential characteristics of this distribution are the sum of probabilities equals one, and each probability value lies between zero and one.
The probable question may be:
Imagine a village where the number of books in the library grows over the years. Let's use a function to represent this: f(t), where t is the number of years elapsed. The function f(t) tells us how many books are in the library at any given time.
Additional Information:
In this story, t is measured in years, and f(t) is the number of books in the library. Now, we want to explore the domain and range of this function.
Domain (Input):
The domain represents the possible values for t (years). As our village is flourishing, let's consider a reasonable range of t from 0 to 50 years. So, the domain is 0≤t≤50
Range (Output):
The range signifies the possible values for f(t) (number of books). Assuming the library is steadily growing, the range could be from the current number of books, let's say 500, to an optimistic estimate of 10,000 books. Therefore, the range is 500≤f(t)≤10,000