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The image shows two problems involving the calculation of the expected value \(E(X)\) for a random variable \(X\) given its probability distribution.

### Problem 1:
The random variable \(X\) and its probability distribution \(P(X = x)\) are given in the table:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -9 & -7 & -5 & -3 & -1 & 1 \\
\hline
P(X = x) & 0.17 & 0.18 & 0.13 & 0.11 & 0.14 & 0.27 \\
\hline
\end{array}
\]

The expected value is calculated using the formula:

\[
E(X) = \sum (x \cdot P(X = x))
\]

### Calculation:
\[
E(X) = (-9)(0.17) + (-7)(0.18) + (-5)(0.13) + (-3)(0.11) + (-1)(0.14) + (1)(0.27)
\]

\[
E(X) = -1.53 + (-1.26) + (-0.65) + (-0.33) + (-0.14) + 0.27
\]

\[
E(X) = -3.64
\]

So, the expected value \(E(X) \approx -3.64\).

---

### Problem 2:
For the second problem, the table provides the following dataThe image shows two problems involving the calculation of the expected value \(E(X)\) for a random variable \(X\) given its probability distribution.

### Problem 1:
The random variable \(X\) and its probability distribution \(P(X = x)\) are given in the table:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -9 & -7 & -5 & -3 & -1 & 1 \\
\hline
P(X = x) & 0.17 & 0.18 & 0.13 & 0.11 & 0.14 & 0.27 \\
\hline
\end{array}
\]

The expected value is calculated using the formula:

\[
E(X) = \sum (x \cdot P(X = x))
\]

### Calculation:
\[
E(X) = (-9)(0.17) + (-7)(0.18) + (-5)(0.13) + (-3)(0.11) + (-1)(0.14) + (1)(0.27)
\]

\[
E(X) = -1.53 + (-1.26) + (-0.65) + (-0.33) + (-0.14) + 0.27
\]

\[
E(X) = -3.64
\]

So, the expected value \(E(X) \approx -3.64\).

---

### Problem 2:
For the second problem, the table provides the following 

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & 3 & 6 & 9 & 12 & 15 \\
\hline
P(X = x) & \frac{1}{8} & \frac{1}{8} & \frac{1}{4} & \frac{1}{4} & \frac{1}{8} & \frac{1}{8} \\
\hline
\end{array}
\]

Using the same formula for \(E(X)\):

\[
E(X) = \sum (x \cdot P(X = x))
\]

### Calculation:
\[
E(X) = (0)(\frac{1}{8}) + (3)(\frac{1}{8}) + (6)(\frac{1}{4}) + (9)(\frac{1}{4}) + (12)(\frac{1}{8}) + (15)(\frac{1}{8})
\]

\[
E(X) = 0 + 0.375 + 1.5 + 2.25 + 1.5 + 1.875
\]

\[
E(X) = 7.5
\]

Thus, the expected value \(E(X) = 7.5\).

---

Would you like more detailed steps or explanations?

Here are 5 follow-up questions for deeper exploration:
1. How does the concept of expected value relate to the mean of a probability distribution?
2. Can you explain how changing probabilities affects the expected value?
3. What would happen to the expected value if all probabilities were equal?
4. How does the expected value differ from the median in a probability distribution?
5. Can you calculate the variance for these probability distributions?

**Tip:** When calculating expected values, always ensure that the sum of probabilities equals 1, as this is a fundamental property of probability distributions.

Find the expected value E(X) of a random variable X having the following probability distribution.

Learn how to calculate the expected value E(X) for a random variable X using a probability distribution. The problem involves multiplying each value of X by its probability and summing the results to find the expected value. Suitable for high school to college-level statistics students.

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