- The mean of a discrete random variable is the sum of each value multiplied by its probability: \(\mu = \sum x_i P(x_i)\).
- Multiply each study time \(x_i\) by its corresponding probability \(P(x_i)\) to get: \(0.035, 0.2, 0.69, 0.4\), and \(0.175\).
- Sum these values together to determine the mean: \(0.035 + 0.2 + 0.69 + 0.4 + 0.175 = 1.5\).
- The calculated mean of the distribution is \(\boxed{1.5}\).
Explanation
- Analyze the probability distribution and recall the formula for the mean
The problem asks for the mean, or expected value, of a discrete random variable \(X\), which represents the number of hours students spend studying. The mean \(\mu\) is calculated using the formula \(\mu = E[X] = \sum x_i P(x_i)\), where \(x_i\) is each possible value of the random variable and \(P(x_i)\) is its corresponding probability. We are given the following distribution:
- \(x_1 = 0.5, P(x_1) = 0.07\)
- \(x_2 = 1, P(x_2) = 0.2\)
- \(x_3 = 1.5, P(x_3) = 0.46\)
- \(x_4 = 2, P(x_4) = 0.2\)
- \(x_5 = 2.5, P(x_5) = 0.07\)
- Calculate the products for each variable value
To find the expected value, we calculate the product of each study duration \(x_i\) and its probability \(P(x_i)\):
- For \(x_1 = 0.5\): \(0.5 \times 0.07 = 0.035\)
- For \(x_2 = 1\): \(1 \times 0.2 = 0.2\)
- For \(x_3 = 1.5\): \(1.5 \times 0.46 = 0.69\)
- For \(x_4 = 2\): \(2 \times 0.2 = 0.4\)
- For \(x_5 = 2.5\): \(2.5 \times 0.07 = 0.175\)
- Sum the products to find the mean
Finally, we sum these individual products to obtain the mean \(\mu\):
\(\mu = 0.035 + 0.2 + 0.69 + 0.4 + 0.175\)
Adding these values step-by-step:
\(0.035 + 0.175 = 0.21\)
\(0.2 + 0.4 = 0.6\)
\(\mu = 0.21 + 0.69 + 0.6 = 0.9 + 0.6 = 1.5\)
Thus, the mean of the probability distribution is \(1.5\) hours. Note: The provided multiple-choice options do not include the correct calculated value.
Examples
Calculating the mean is crucial in real-life data analysis, such as evaluating student performance or predicting business outcomes. If you are a teacher studying study habits, you can use the expected value to understand the "average" time students commit to learning, which helps in planning your curriculum load. The mean here is computed as \(\mu = \sum x_i P(x_i) = \boxed{1.5} \text{ hours}\).