- To find the expected age, multiply each age by its probability and sum the results.
- The formula is: \(E[X] = \sum x_i P(X = x_i)\).
- Performing the calculation: \(6(0.5) + 7(0.1) + 8(0.2) + 9(0.2) = 3.0 + 0.7 + 1.6 + 1.8 = 7.1\).
- Final answer: \(\boxed{7.1 \text{ years}}\)
Explanation
- Understanding the Expected Value Formula and Data Analysis
To find the expected age of a passenger on the train, we use the definition of the expected value of a discrete random variable. The expected value, denoted by \(E[X]\), represents the long-term average value of the variable. It is calculated by summing the products of each possible age \(x_i\) and its corresponding probability \(P(X = x_i)\). The formula is given by:
\[E[X] = \sum_{i=1}^{n} x_i \cdot P(X=x_i)\]
Given the ages \(X \in \{6, 7, 8, 9\}\) and their associated probabilities \(P(X=6) = \frac{1}{2}\), \(P(X=7) = \frac{1}{10}\), \(P(X=8) = \frac{1}{5}\), and \(P(X=9) = \frac{1}{5}\), we first ensure these probabilities sum to \(1\):
\[\frac{1}{2} + \frac{1}{10} + \frac{1}{5} + \frac{1}{5} = \frac{5}{10} + \frac{1}{10} + \frac{2}{10} + \frac{2}{10} = \frac{10}{10} = 1\]
Since the sum is equal to \(1\), this is a valid probability distribution.
- Calculating the Expected Value
Now, we substitute the given values into the expected value formula:
\[E[X] = (6 \times P(X=6)) + (7 \times P(X=7)) + (8 \times P(X=8)) + (9 \times P(X=9))\]
Substituting the probabilities as decimals (\(\frac{1}{2}=0.5\), \(\frac{1}{10}=0.1\), \(\frac{1}{5}=0.2\)):
\[E[X] = (6 \times 0.5) + (7 \times 0.1) + (8 \times 0.2) + (9 \times 0.2)\]
Now, we perform the multiplication for each term:
- \(6 \times 0.5 = 3.0\)
- \(7 \times 0.1 = 0.7\)
- \(8 \times 0.2 = 1.6\)
- \(9 \times 0.2 = 1.8\)
Finally, sum these values to find the expected age:
\[E[X] = 3.0 + 0.7 + 1.6 + 1.8 = 7.1\]
The expected age for one of the passengers is \(7.1\) years.
Examples
In real life, expected value is a powerful tool used in insurance and finance to predict average outcomes. For instance, an insurance company uses the expected value of claim costs to set reasonable premiums. If they know that in a group of 100 people, 90 will not make a claim (cost = \$0) and 10 will make a \$1,000 claim, the expected cost per person is \(0.9 \times 0 + 0.1 \times 1000 = \$100\). By knowing this average, they can ensure their business remains sustainable while providing coverage.