Find the mean of the random variable x described in the preceding exercise.

Verified step by step guidance

1. Step 1: Recall the formula for the mean (expected value) of a random variable x. The mean is calculated as \( \mu = \sum_{i} x_i P(x_i) \), where \( x_i \) represents the values of the random variable and \( P(x_i) \) represents their corresponding probabilities.
2. Step 2: Identify the values of \( x_i \) (the possible outcomes of the random variable) and \( P(x_i) \) (the probabilities associated with each outcome) from the preceding exercise.
3. Step 3: Multiply each value \( x_i \) by its corresponding probability \( P(x_i) \). This gives the weighted contribution of each outcome to the mean.
4. Step 4: Sum all the weighted contributions \( \sum_{i} x_i P(x_i) \) to compute the mean of the random variable.
5. Step 5: Ensure that the probabilities \( P(x_i) \) sum to 1, as this is a requirement for a valid probability distribution. If they do not, revisit the preceding exercise to verify the probabilities.

Key Concepts

Mean

The mean, often referred to as the average, is a measure of central tendency that summarizes a set of values by dividing the sum of all values by the number of values. In the context of a random variable, the mean represents the expected value, indicating where the center of the distribution lies. It is a crucial concept in statistics as it provides a single value that represents the entire dataset.

Random Variable

A random variable is a numerical outcome of a random phenomenon, which can take on different values based on the outcome of a random event. Random variables can be discrete, taking on specific values, or continuous, taking on any value within a range. Understanding random variables is essential for calculating probabilities and statistical measures like the mean.

Expected Value

The expected value of a random variable is a fundamental concept in probability and statistics that represents the long-term average of the variable's outcomes. It is calculated by multiplying each possible outcome by its probability and summing these products. The expected value provides insight into the average result one can anticipate from a random variable over numerous trials.