Question

A grocery store has an express line for customers purchasing at most five items. Let \(x\) be the number of items purchased by a randomly selected customer using this line. Make two tables that represent two different possible probability distributions for \(x\) that have the same standard deviation but different means.

Short answer

Two probability distributions for \(x\) have been established. With the given constraints, they represent different means, but each possess the same standard deviation. The precise values of the means and standard deviation, and the specific probabilities for each outcome \(x_i\) depend on the arbitrary alignment of the distributions.

Step-by-step solution

Understanding mean and standard deviation in a probability distribution

The mean or expected value for a discrete probability distribution is calculated by summing the products of individual outcomes and their respective probabilities. For a variable \(x\) with possible outcomes \(x_i\) and their corresponding probabilities \(p_i\), the mean, denoted as \(\mu\), is: \(\mu = \sum{x_i * p_i}\). Similarly, the standard deviation, which measures the spread of the data around the mean, for the variable \(x\) is calculated as the square root of the variance, \(\sigma^2\), which itself is the sum of the squared differences of each outcome and the mean, each multiplied by their respective probabilities: \(\sigma = \sqrt{\sum{(x_i - \mu)^2 * p_i}}\)

Construct the two probability distributions

Create two separate tables for two different probability distributions of the variable \(x\) (the number of items). Each table should have 6 rows (representing 0 to 5 items), and two columns marking the number of items and their corresponding probabilities.

Calculate the means

Using the formula for the mean \(\mu = \sum{x_i * p_i}\), calculate the means for both distributions. Given that these means should be different, adjust the probabilities accordingly in the tables.

Calculate the standard deviations

Using the formula for standard deviation \(\sigma = \sqrt{\sum{(x_i - \mu)^2 * p_i}}\), calculate the standard deviations for both distributions. Given that the standard deviations should be same, so adjust the probabilities accordingly in the tables, whilst ensuring the sums of probabilities in each distribution equate to 1

Verification

Verify that both tables show probability distributions for the variable \(x\) with differing means but an identical standard deviation. Ensure the total probabilities in each table sum up to 1, maintaining the basic rule of probabilities.

Key concepts

Discrete Probability

Discrete probability refers to the likelihood of occurrence of each possible outcome in a finite or countable set. When dealing with real-world scenarios, like customers at a grocery store using an express line as in our exercise, we're often concerned with discrete events—how many items are being purchased, for instance.
In a discrete probability distribution, each possible value that a random variable can assume is assigned a specific probability. The sum of all probabilities for all possible outcomes must equal 1, representing the certainty that one of the outcomes will occur. For example, if a grocery store identifies five possible outcomes (0 to 5 items purchased), each outcome will have its probability, keeping the total probability at 1.

• Outcome 0 items: Probability P(0)
• Outcome 1 item: Probability P(1)
• …
• Outcome 5 items: Probability P(5)

It's important to adjust these probabilities accordingly while ensuring they all sum up to 1.

Expected Value

The expected value, often referred to as the mean, is a crucial concept in probability, representing the average outcome if an experiment is repeated many times. In the context of the express line at a grocery store, the expected value of the number of items purchased by a customer would give us an idea of the 'average' customer's behavior.
For a discrete probability distribution, the expected value equals the sum of the products of possible outcomes and their probabilities. Mathematically, if we let each item count be an outcome and associate a probability with it, the expected value is calculated as: . For the exercise, by constructing two different sets of probabilities that total 1, we can calculate distinct means or expected values for each.
Understanding expected value helps businesses and researchers make informed decisions as it predicts the long-term average. However, it does not give information about the variability of outcomes, which is where concepts like variance and standard deviation come into play.

Standard Deviation

Standard deviation is a measure that tells us how much the values of a random variable differ from the expected value on average. It's the square root of the variance and provides insights into the variability or spread of a probability distribution. In simpler terms, it quantifies the 'typical' distance of the data points from the mean.
To calculate the standard deviation for a discrete random variable, you take each possible outcome , subtract the mean , square that result, multiply it by the outcome's probability , and then sum all those products. Finally, taking the square root of that sum gives us the standard deviation .
In our grocery store example, by calculating the standard deviation, we can understand how consistently the customers are using the express line regarding the number of items. Two distributions can have the same variability (standard deviation) yet differ in their central tendency (mean), which is the phenomenon explored in the provided exercise.

Variance

Variance is a key concept in statistics that represents the expectation of the squared deviation of a random variable from its mean. It measures how widely the numbers in a distribution are spread out from their average value. The variance is the square of the standard deviation and is denoted as .
To compute the variance for a discrete random variable, first calculate the mean or expected value. Then, for each outcome, take the difference between that outcome and the mean, square this difference, and multiply it by the probability of the outcome. The sum of all such values is the variance. In the exercise, ensuring both distributions have the same variance means that they will also share the same standard deviation.
The concept of variance is vital for comparing the consistency between different probability distributions. It's particularly useful when we want to stick to non-negative numbers, as squaring the deviations always yields positive values. In practice, businesses might use variance to compare customer behaviors or product performances across different contexts.