Expected Value E(X) Calculation With Probability Distribution

Hey guys! Today, we're diving into the fascinating world of probability distributions and how to calculate the expected value, often denoted as E(X). This is a super important concept in statistics and probability, and it helps us understand what the average outcome of a random variable will be over the long run. So, let's break it down in a way that's easy to grasp.

Understanding Expected Value

First off, what exactly is the expected value? Simply put, it's the weighted average of all possible values of a random variable, where the weights are the probabilities of observing those values. Think of it as the average outcome you'd expect if you repeated an experiment many, many times. It's a crucial concept in fields like finance, gambling, and even weather forecasting. In finance, for instance, the expected value helps investors assess the potential return on investment, considering the probabilities of different market scenarios. A higher expected value suggests a more promising investment, while a lower one signals higher risk or a less attractive opportunity. In gambling, the expected value is used to determine whether a game of chance is fair or advantageous to the player or the house. A negative expected value implies that the player is likely to lose money in the long run, while a positive one indicates a potential advantage. Even in weather forecasting, expected values are used to predict the average amount of rainfall or the average temperature over a given period, based on historical data and probability models. Understanding the expected value helps in making informed decisions and planning for the future.

To really nail this, let's think about a simple example: imagine you're flipping a coin. There are two possible outcomes: heads or tails. If the coin is fair, the probability of each outcome is 0.5 (or 50%). If we assign a value of 1 to heads and 0 to tails, the expected value would be (1 * 0.5) + (0 * 0.5) = 0.5. This means that, on average, you'd expect to get heads half the time if you flipped the coin many times. This might seem straightforward, but it lays the foundation for understanding more complex scenarios with multiple outcomes and different probabilities. The expected value isn't just a theoretical concept; it's a practical tool that helps us make sense of the world around us and make better decisions in the face of uncertainty. So, with this basic understanding in place, let's move on to tackling the specific problem we have at hand – calculating the E(X) from a given probability distribution table. We'll walk through the steps together, making sure you're confident in applying this knowledge to any similar problem you encounter.

Setting up the Problem

Okay, so we have a discrete random variable X, and we’re given its probability distribution in a table. A discrete random variable is simply a variable whose value can only take on a finite number of values or a countably infinite number of values. Think of it like counting things – you can have 1, 2, 3 items, but you can't have 2.5 items. In our case, X can take on the values 3, 4, 5, 6, and 7. Each of these values has an associated probability, denoted as P(X = x). The probability distribution tells us how likely each value of X is to occur. This is super useful because it gives us a complete picture of the possible outcomes and their likelihoods. For instance, if we were dealing with the number of heads obtained from flipping a coin three times, the possible values for X would be 0, 1, 2, or 3 heads. Each of these outcomes would have a specific probability, depending on the fairness of the coin. Similarly, if X represented the number of customers entering a store in an hour, it could take on integer values like 0, 1, 2, and so on. The probability distribution would tell us how often we might expect to see each of these customer counts. Understanding discrete random variables and their probability distributions is essential for calculating the expected value, as it provides the framework for weighing each possible outcome by its probability of occurrence. This process helps us to arrive at a meaningful average value that represents the long-term expectation of the random variable. So, with this concept in mind, let's now turn to the specifics of our problem and see how we can use the given data to calculate E(X).

Here’s the table we're working with:

Our goal is to find E(X), which represents the expected value of the random variable X. Remember, the expected value is essentially the long-run average value we would expect to see if we repeated the experiment many times. It's like predicting the average grade a student will get over many tests or the average number of customers a store will see each day. To calculate the expected value, we need to use a specific formula that takes into account each possible value of the random variable and its corresponding probability. This formula ensures that each outcome is weighted appropriately, giving more importance to those outcomes that are more likely to occur. Before we jump into the calculations, it's worth noting that the sum of all probabilities in the distribution should always equal 1. This makes sense because the probabilities represent the likelihood of all possible outcomes, and one of those outcomes must occur. So, as a quick check, we can add up the probabilities in our table: 0.3 + 0.2 + 0.2 + 0.1 + 0.2 = 1. This confirms that we have a valid probability distribution, and we can confidently proceed with calculating the expected value. Now, let's get to the formula and see how we can apply it to our specific problem.

The Formula for Expected Value

The formula for the expected value of a discrete random variable X is:

E(X) = Σ [x * P(X = x)]

Where:

• E(X) is the expected value of X
• Σ means we need to sum up the values
• x represents each possible value of the random variable
• P(X = x) is the probability of X taking on the value x

This formula might look a bit intimidating at first, but it's actually quite straightforward once you break it down. The Σ symbol, which is the Greek letter sigma, simply tells us that we need to add up a series of terms. In this case, each term is the product of a possible value of the random variable (x) and its corresponding probability (P(X = x)). So, what we're doing is multiplying each outcome by its probability and then summing up all those products. This gives us a weighted average, where the weights are the probabilities. To illustrate this further, let's go back to our coin-flipping example. If we were calculating the expected value of the number of heads obtained in one flip, we would multiply the value for heads (1) by its probability (0.5) and add it to the value for tails (0) multiplied by its probability (0.5). This would give us (1 * 0.5) + (0 * 0.5) = 0.5, as we saw earlier. Similarly, for more complex scenarios with multiple outcomes, we just extend this process by adding up the products for each outcome. The key to using this formula effectively is to ensure that you correctly identify all the possible values of the random variable and their corresponding probabilities. Once you have this information, it's just a matter of plugging the values into the formula and performing the calculations. So, with a clear understanding of the formula, let's now apply it to our specific problem and see how we can find the expected value E(X) from the given probability distribution table.

Calculating E(X)

Alright, let's put that formula into action! To calculate E(X), we need to multiply each value of x by its corresponding probability P(X = x) and then add them all up. Here’s how it looks for our table:

E(X) = (3 * 0.3) + (4 * 0.2) + (5 * 0.2) + (6 * 0.1) + (7 * 0.2)

Now, let’s break it down step-by-step:

• 3 * 0.3 = 0.9
• 4 * 0.2 = 0.8
• 5 * 0.2 = 1.0
• 6 * 0.1 = 0.6
• 7 * 0.2 = 1.4

Now, we add these products together:

E(X) = 0.9 + 0.8 + 1.0 + 0.6 + 1.4 = 4.7

So, the expected value E(X) is 4.7. This means that, on average, if we were to repeat the experiment associated with this probability distribution many times, we would expect the outcome to be around 4.7. The calculation process might seem a bit tedious at first, especially if you're dealing with a large number of possible values. However, it's a very systematic process that just involves multiplying and adding. To help you visualize this, think of each value of x as a possible score in a game, and P(X = x) as the probability of achieving that score. The expected value is then like the average score you would expect to get if you played the game many times. It's a powerful concept because it gives you a single number that summarizes the overall tendency of the random variable. In our case, 4.7 tells us that the values of X are centered around this number, taking into account their respective probabilities. Now, let's move on to the final step, which is rounding our answer to the required decimal place and presenting the final result. This will ensure that our answer is in the correct format and easy to interpret.

Rounding the Answer

The question asks us to round our answer to one decimal place, and guess what? We already have our answer in that format! E(X) = 4.7. So, there's no extra rounding needed in this case. Sometimes, you might get an answer with more decimal places, and you'll need to round it appropriately. For example, if we had gotten 4.73, we would round it down to 4.7, and if we had gotten 4.78, we would round it up to 4.8. Rounding is a crucial step in many calculations because it helps to simplify the answer and make it easier to understand. It also reflects the level of precision that is appropriate for the given context. In scientific and engineering applications, for instance, the number of decimal places often indicates the accuracy of the measurement. Similarly, in financial calculations, rounding to the nearest cent is a common practice. In our case, rounding to one decimal place is sufficient because it provides a clear and concise representation of the expected value. The key is to follow the instructions given in the question and to use standard rounding rules, which state that if the next digit is 5 or greater, you round up, and if it's less than 5, you round down. Now that we've ensured our answer is in the correct format, let's present the final answer and recap what we've learned in this problem.

Final Answer

The expected value E(X) is 4.7.

So, there you have it! We’ve successfully calculated the expected value E(X) from the given probability distribution table. Remember, this value represents the average outcome we’d expect over the long run. It's like finding the center of gravity for the distribution, where the probabilities act as weights. We started by understanding what expected value means and how it's a crucial concept in probability and statistics. Then, we identified the given probability distribution and set up the problem. We used the formula E(X) = Σ [x * P(X = x)] to calculate the expected value, which involved multiplying each value by its probability and summing the results. We then rounded our answer to one decimal place, as requested, and arrived at the final answer of 4.7. This process might seem complex at first, but with practice, it becomes second nature. The key is to break the problem down into smaller steps, understand the underlying concepts, and apply the formula systematically. Each step we took, from understanding the formula to the final calculation, built upon the previous one, leading us to the correct solution. The power of expected value lies in its ability to provide a single, meaningful number that summarizes the overall tendency of a random variable. With this understanding, you'll be able to tackle similar problems with confidence and apply the concept of expected value in various real-world scenarios. Great job, guys! You nailed it!