To find the expected value $\mu$ of a discrete random variable $X$, we use the formula:

$\mu = E(X) = \sum \big(x \cdot P(X = x)\big)$

Here, each value of $X = x$ is multiplied by its corresponding probability $P(X = x)$, and the results are summed up.

From the table:

Now, compute $x \cdot P(X = x)$ for each pair and sum them:

$E(X) = 2(0.07) + 4(0.19) + 6(0.25) + 8(0.11) + 10(0.07) + 12(0.30) + 14(0.01)$

Let's calculate this. Values of x and their corresponding probabilities x_values = [2, 4, 6, 8, 10, 12, 14] probabilities = [0.07, 0.19, 0.25, 0.11, 0.07, 0.30, 0.01]

Calculating the expected value

expected_value = sum(x * p for x, p in zip(x_values, probabilities)) expected_valueThe expected value $\mu$ of the probability distribution is:

$\mu = 7.72$

The correct option is:

$\mu = 7.72$.

Would you like further explanations, or have any additional questions about this topic?

Here are 5 related questions to explore further:

1. How do you calculate the variance of a discrete random variable?
2. What does the expected value of a random variable represent in probability theory?
3. How is the expected value applied in real-life scenarios like finance or insurance?
4. What is the relationship between expected value and the mean of a dataset?
5. How would the probabilities or outcomes being incorrect affect the expected value?

Tip: Always ensure the probabilities in a discrete probability distribution add up to 1. This helps verify correctness!