Analyzing Random Variable X And Relative Frequencies In Bookstore Sales
In the realm of probability and statistics, understanding random variables and their distributions is crucial for analyzing real-world phenomena. This article delves into a scenario involving a bookstore that sells books at three different prices: $2, $3, and $5. We will explore the concept of a random variable X, which represents the amount of money a customer spends on a single book. Furthermore, we will analyze a relative frequency table that illustrates the probability distribution of this random variable. By examining the relative frequencies, we can gain insights into customer purchasing patterns and the overall revenue generation for the bookstore.
Defining the Random Variable X
In this context, the random variable X is defined as the amount of money a customer spends on one book. Since the bookstore sells books at three distinct prices, the random variable X can take on three possible values: $2, $3, or $5. Each of these values corresponds to a specific outcome, namely, the purchase of a book at that particular price point. It is important to note that X is a discrete random variable because it can only assume a finite number of values.
Constructing the Relative Frequency Table
The relative frequency table is a powerful tool for summarizing the distribution of a discrete random variable. It displays the possible values of the random variable along with their corresponding relative frequencies, which represent the proportion of times each value occurs in a given dataset. In our bookstore scenario, the relative frequency table would look like this:
| Amount of Money (X) | Relative Frequency |
|---|---|
| $2 | P(X = 2) |
| $3 | P(X = 3) |
| $5 | P(X = 5) |
Where P(X = 2) represents the relative frequency of customers purchasing a book for $2, P(X = 3) represents the relative frequency of customers purchasing a book for $3, and P(X = 5) represents the relative frequency of customers purchasing a book for $5. The sum of all relative frequencies must equal 1, reflecting the fact that all possible outcomes are accounted for.
Interpreting the Relative Frequencies
The relative frequencies provide valuable information about customer purchasing behavior. For instance, if P(X = 2) is the highest relative frequency, it suggests that the majority of customers prefer to buy books priced at $2. Conversely, if P(X = 5) is the lowest relative frequency, it indicates that fewer customers opt for the more expensive books. This information can be used by the bookstore management to make informed decisions regarding pricing strategies, inventory management, and marketing campaigns.
Calculating Expected Value and Variance
With the relative frequency table at hand, we can calculate two important measures that characterize the distribution of the random variable X: the expected value and the variance.
Expected Value
The expected value, denoted as E(X), represents the average amount of money a customer is expected to spend on a book. It is calculated by multiplying each possible value of X by its corresponding relative frequency and summing the products. Mathematically, the expected value is given by:
E(X) = Σ [X * P(X)]
In our bookstore example, the expected value would be calculated as:
E(X) = ($2 * P(X = 2)) + ($3 * P(X = 3)) + ($5 * P(X = 5))
The expected value provides a central tendency measure for the distribution of X. It indicates the typical amount a customer spends on a book, on average.
Variance
The variance, denoted as Var(X), measures the spread or dispersion of the distribution of X around its expected value. It quantifies how much the individual values of X deviate from the average. The variance is calculated by first finding the squared difference between each possible value of X and the expected value, then multiplying each squared difference by its corresponding relative frequency, and finally summing the products. Mathematically, the variance is given by:
Var(X) = Σ [(X - E(X))^2 * P(X)]
In our bookstore scenario, the variance would be calculated as:
Var(X) = [($2 - E(X))^2 * P(X = 2)] + [($3 - E(X))^2 * P(X = 3)] + [($5 - E(X))^2 * P(X = 5)]
The variance provides a measure of the variability in customer spending. A higher variance indicates that customer spending is more spread out, while a lower variance suggests that customer spending is more concentrated around the expected value.
Standard Deviation
Another useful measure of dispersion is the standard deviation, which is simply the square root of the variance. The standard deviation, denoted as SD(X), is expressed in the same units as the random variable X, making it easier to interpret than the variance. Mathematically, the standard deviation is given by:
SD(X) = √Var(X)
In our bookstore example, the standard deviation would be calculated by taking the square root of the variance calculated earlier.
The standard deviation provides a more intuitive measure of the spread of the distribution than the variance. It represents the typical deviation of customer spending from the average.
Applications and Implications
The analysis of the random variable X and its relative frequency distribution has several practical applications for the bookstore management. By understanding customer purchasing patterns, the bookstore can:
- Optimize Pricing Strategies: The bookstore can adjust its pricing strategy based on the relative frequencies of different price points. For example, if books priced at $2 are the most popular, the bookstore might consider offering more books in that price range or running promotions to further incentivize purchases.
- Manage Inventory Effectively: The bookstore can use the relative frequencies to forecast demand for books at different price points. This information can help the bookstore manage its inventory levels, ensuring that it has enough books in stock to meet customer demand without overstocking less popular items.
- Target Marketing Campaigns: The bookstore can tailor its marketing campaigns to specific customer segments based on their purchasing preferences. For instance, if customers who buy $5 books are more likely to be interested in a particular genre, the bookstore can target them with promotions for new releases in that genre.
- Estimate Revenue and Profit: The bookstore can use the expected value of X to estimate its average revenue per customer. This information can be used to project overall revenue and profit, helping the bookstore make informed financial decisions.
Example Scenario and Calculations
To illustrate the concepts discussed above, let's consider a specific example. Suppose the relative frequency table for the bookstore is as follows:
| Amount of Money (X) | Relative Frequency (P(X)) |
|---|---|
| $2 | 0.4 |
| $3 | 0.3 |
| $5 | 0.3 |
This table indicates that 40% of customers buy books priced at $2, 30% buy books priced at $3, and 30% buy books priced at $5.
Calculating Expected Value
The expected value of X can be calculated as follows:
E(X) = ($2 * 0.4) + ($3 * 0.3) + ($5 * 0.3) = $0.8 + $0.9 + $1.5 = $3.2
This means that, on average, a customer is expected to spend $3.20 on a book at this bookstore.
Calculating Variance
The variance of X can be calculated as follows:
Var(X) = [($2 - $3.2)^2 * 0.4] + [($3 - $3.2)^2 * 0.3] + [($5 - $3.2)^2 * 0.3]
= [(-$1.2)^2 * 0.4] + [(-$0.2)^2 * 0.3] + [($1.8)^2 * 0.3]
= (1.44 * 0.4) + (0.04 * 0.3) + (3.24 * 0.3)
= 0.576 + 0.012 + 0.972 = 1.56
This indicates that the variance of customer spending is 1.56 squared dollars.
Calculating Standard Deviation
The standard deviation of X can be calculated as follows:
SD(X) = √1.56 ≈ $1.25
This means that the typical deviation of customer spending from the average of $3.20 is approximately $1.25.
Conclusion
In conclusion, the analysis of the random variable X, representing the amount of money spent on a book, and its relative frequency distribution provides valuable insights into customer purchasing behavior and the financial performance of the bookstore. By constructing and interpreting the relative frequency table, calculating the expected value, variance, and standard deviation, the bookstore management can make informed decisions regarding pricing, inventory, marketing, and overall business strategy. This example highlights the power of probability and statistics in understanding and optimizing real-world scenarios.
Original Question: A bookstore sells books for $2, $3, and $5. Let random variable X = 'amount of money for one book'. Look at the relative-frequency table below representing the amount of money spent on one item and the relative frequencies with which customers purchase. Discussion category: mathematics.
Rewritten Question: A bookstore offers books at prices of $2, $3, and $5. Define a random variable X as the price of a book purchased by a customer. Given a relative frequency table showing the distribution of book prices, analyze the customer purchasing patterns and calculate relevant statistical measures. How can this information assist the bookstore in making strategic decisions regarding pricing, inventory, and marketing?
