Let X be a random variable with the following probability distribution.

Value x of X

P(X=x)

-2

0.15

-1

0.20

0

0.10

1

0.10

2

0.45

Complete the following. (if necessary, consult a list of formulas.)

(9) Find the expectation E(X) l X. E(X)=

(b) Find the variance Var(X) of X var(x)=

Given the probability distribution of the random variable \( X \), we can calculate the expectation \( E(X) \) and the variance \( \text{Var}(X) \).

### Probability Distribution:
| Value \( x \) of \( X \) | \( P(X = x) \) |
|:------------------------:|:--------------:|
| -2                       | 0.15           |
| -1                       | 0.20           |
|  0                       | 0.10           |
|  1                       | 0.10           |
|  2                       | 0.45           |

### (a) Expectation \( E(X) \):
The expectation \( E(X) \) is calculated using the formula:
\[
E(X) = \sum_{i} x_i \cdot P(X = x_i)
\]
Substituting the values:
\[
E(X) = (-2 \times 0.15) + (-1 \times 0.20) + (0 \times 0.10) + (1 \times 0.10) + (2 \times 0.45)
\]
\[
E(X) = -0.30 + (-0.20) + 0 + 0.10 + 0.90 = 0.50
\]
So, \( E(X) = 0.50 \).

### (b) Variance \( \text{Var}(X) \):
The variance \( \text{Var}(X) \) is calculated using the formula:
\[
\text{Var}(X) = E(X^2) - [E(X)]^2
\]
First, we calculate \( E(X^2) \):
\[
E(X^2) = \sum_{i} x_i^2 \cdot P(X = x_i)
\]
Substituting the values:
\[
E(X^2) = (-2)^2 \times 0.15 + (-1)^2 \times 0.20 + 0^2 \times 0.10 + 1^2 \times 0.10 + 2^2 \times 0.45
\]
\[
E(X^2) = 4 \times 0.15 + 1 \times 0.20 + 0 \times 0.10 + 1 \times 0.10 + 4 \times 0.45 = 0.60 + 0.20 + 0 + 0.10 + 1.80 = 2.70
\]

Now, using \( E(X) = 0.50 \):
\[
\text{Var}(X) = E(X^2) - [E(X)]^2 = 2.70 - (0.50)^2 = 2.70 - 0.25 = 2.45
\]

### Final Answers:
- \( E(X) = 0.50 \)
- \( \text{Var}(X) = 2.45 \)

Would you like more details or have any questions?

### Related Questions:
1. How does the sign of the values in a probability distribution affect the expectation?
2. What is the relationship between the expectation and the symmetry of a distribution?
3. How would adding a constant to all values of \( X \) affect the expectation and variance?
4. How does the variance reflect the spread of negative and positive values in the distribution?
5. Can the variance ever be negative? Why or why not?
6. What is the standard deviation of \( X \) given the variance \( \text{Var}(X) \)?
7. How does changing the probabilities in the distribution affect the variance?
8. What is the skewness of this distribution, and how does it relate to the expectation and variance?

### Tip:
When dealing with a distribution that includes negative values, the expectation can indicate whether the distribution is skewed towards the negative or positive side.

Learn how to calculate the expected value (E(X)) and variance (Var(X)) of a random variable X with a given probability distribution. This problem involves understanding probability theory concepts and applying the formulas for expected value and variance. Suitable for undergraduate-level mathematics.

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