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 \(x\) : $$\begin{array}{l|cccc}x & 0 & 1 & 2 & 3 \\\\\hline p(x) & .1 & .4 & .4 & .1\end{array}$$ Find the expected value of \(x\), the average number of times a customer visits the store.

Short answer

Answer: The expected number of times a customer visits the grocery store in a 1-week period is 1.5 visits per week.

Step-by-step solution

List the given probabilities and values of x

We are given that the probability distribution table for x is: $$\begin{array}{l|cccc}x & 0 & 1 & 2 & 3 \\\\\hline p(x) & .1 & .4 & .4 & .1\end{array}$$ Step 2:

Multiply the probabilities with their respective values of x

We need to multiply each probability by the corresponding value of x to get the product. Calculate these products as shown below: $$ (0 \times 0.1) = 0 \\ (1 \times 0.4) = 0.4 \\ (2 \times 0.4) = 0.8 \\ (3 \times 0.1) = 0.3 $$ Step 3:

Calculate the expected value

Using the formula for the expected value of a discrete random variable, add up the products calculated in Step 2 to find the expected value: $$ E[X] = 0 + 0.4 + 0.8 + 0.3 = 1.5 $$ Thus, the expected value of \(x\) (the average number of times a customer visits the grocery store in a 1-week period) is 1.5 visits per week.

Key concepts

Probability Distribution

Imagine flipping a coin. You anticipate either heads or tails - an example of a simple probability distribution, which tells us the likelihood of each outcome. However, not all situations are this straightforward. In real-world scenarios, such as determining how often customers visit a grocery store, things get more complex.

Dealing with multiple outcomes, the probability distribution is essentially a table or formula that links each possible outcome of a discrete random variable with its probability of occurrence. Like in our exercise, where the random variable is the number of store visits in a week and the distribution is given as a table, indicating how likely each number of visits is.

For each possible value of the variable (0, 1, 2, 3 visits), there's a probability assigned (.1, .4, .4, .1 respectively). Together, these form the probability distribution and must satisfy two conditions: the sum of probabilities must equal 1, and each probability must be between 0 and 1 inclusive. It's this distribution that allows us to calculate the expected value, guiding businesses in forecasting customer behavior and managing resources efficiently.

Discrete Random Variable

A discrete random variable takes on a countable number of distinct values, much like the beads on a necklace. The values don't have to be sequential or uniform; what's essential is they can be listed out. For instance, the number of times a customer visits a store in a week is a perfect example of this type of variable.

The random variable in our exercise, represented by 'x', assumes the values 0, 1, 2, and 3. These integers represent the distinct and countable outcomes of a customer's weekly visits. Unlike continuous random variables, which can take on any value within a range, discrete ones like 'x' allow us to precisely pinpoint and analyze each potential event — an essential quality for calculating the probability distribution and, by extension, the expected value. Remember, the specificity and countability of discrete random variables make them particularly manageable in probability and statistics.

Average Number of Visits

When we talk about the average number of visits, we're referring to a concept deeply rooted in our daily lives — from gauging how often we frequent a place to assessing market trends. In probability, this average is quantified as the expected value of a discrete random variable.

In the context of our exercise, computing the expected value gives us the average number of times customers visit the grocery store in a week. You take each number of visits, multiply it by the chance of that number occurring, and add up the results — effectively blending all possibilities into a single predictive figure.

The expected value doesn't guarantee that every customer will adhere to this average; rather, it's a mathematical construct to describe the center of the distribution. It serves as an invaluable tool for businesses to plan their inventory, staffing, and marketing strategies around the habitual tendencies of their customers.