Question

Let \(x\) represent the number of times a customer visits a grocery store in a 1 -week period. Assume this is the probability distribution of \(x\): $$\begin{array}{l|llll}x & 0 & 1 & 2 & 3 \\\\\hline p(x) & .1 & .4 & .4 & .1\end{array}$$ Find the expected value of \(x\), the average number of times a customer visits the store.

Short answer

- x = 0 with probability 0.1 - x = 1 with probability 0.4 - x = 2 with probability 0.4 - x = 3 with probability 0.1 Answer: The expected value (average) of the number of visits by a customer to the grocery store in a 1-week period is 1.5.

Step-by-step solution

Define the expected value formula for a discrete random variable

The expected value (E(x)) of a discrete random variable is defined as: $$E(x) = \sum_{i=1}^{n} x_i p(x_i)$$ where \(x_i\) represents an individual outcome and \(p(x_i)\) represents the probability of that outcome occurring.

Multiply the probability of each outcome by the corresponding value of x

According to the given probability distribution table, we have the values of x and their respective probabilities: - x = 0 with probability 0.1 - x = 1 with probability 0.4 - x = 2 with probability 0.4 - x = 3 with probability 0.1 Now, we multiply each outcome (x) by its corresponding probability (p(x)): 0 * 0.1 = 0, 1 * 0.4 = 0.4, 2 * 0.4 = 0.8, 3 * 0.1 = 0.3

Sum the results from step 2

Now, we sum up the results that we obtained from the multiplication in step 2: 0 + 0.4 + 0.8 + 0.3 = 1.5

Calculate and present the expected value

The expected value (E(x)) is the sum of the results obtained in step 3, which is: E(x) = 1.5 So the expected value (average) of the number of visits by a customer to the grocery store in a 1-week period is 1.5.

Key concepts

Probability Distribution

In the realm of statistics, probability distribution is a fundamental concept. It provides a framework for understanding how probable different outcomes are within a set of possible results. In this particular exercise, we're looking at how often a customer visits a grocery store over a week. Each potential number of visits (0, 1, 2, or 3) is linked with a chance or probability of occurring. This is presented in a table:

• 0 visits with a probability of 0.1
• 1 visit with a probability of 0.4
• 2 visits with a probability of 0.4
• 3 visits with a probability of 0.1

The table showcases a discrete probability distribution. This means the number of visits is counted in whole numbers—not fractions or decimals. In essence, it's a snapshot of the likelihood of different numbers of visits happening. Understanding probability distribution helps us visualize and calculate average expectations, like how often a customer might visit a store each week.

Discrete Random Variable

A discrete random variable is a specific type of variable in probability and statistics. It is called 'discrete' because it can take on distinct and separate values. Think of it as steps on a ladder, where each step is a whole number, without any in-between steps or fractions.
The exercise gives us the number of visits a customer might make to a grocery store in a week. Here, the random variable is the number of visits denoted by "x."
Since it only takes the values 0, 1, 2, and 3, it fits the definition of a discrete random variable. Each value has its own probability, provided by the probability distribution. Such variables are essential when calculating expected values, as they help organize possible outcomes and their probabilities into a coherent mathematical framework.

Average Number of Visits

The expected value tells us the average result we can anticipate over time. When we calculate the expected value of a discrete random variable, like in this exercise, we are essentially trying to find the average number of visits to the grocery store in a week.
To find this, we multiply each possible number of visits by its probability and then sum these products. Specifically, we perform the following calculations:

• 0 visits * 0.1 = 0
• 1 visit * 0.4 = 0.4
• 2 visits * 0.4 = 0.8
• 3 visits * 0.1 = 0.3

Adding these results gives us an expected value of 1.5. This means that, on average, a customer visits the grocery store 1.5 times per week. The concept of expected value is powerful because it provides a single number that summarizes a complex probability distribution, offering insights into typical or usual behavior over time.