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Let X be a random variable with the following probability distribution. (a) Find the expectation E(X) of X. E(X)= (b) Find the variance Var(X) of X. Var(X)

To find the expectation \(E(X)\) and the variance \(\text{Var}(X)\) of the random variable \(X\) given its probability distribution, we can use the following formulas:

1. **Expectation**: \( E(X) = \sum [x_i \times P(X = x_i)] \)
2. **Variance**: \( \text{Var}(X) = E(X^2) - [E(X)]^2 \), where \( E(X^2) = \sum [x_i^2 \times P(X = x_i)] \)

### Given Data:
- \( x_i \) values: 20, 30, 40, 50, 60, 70
- \( P(X = x_i) \) values: 0.25, 0.10, 0.10, 0.10, 0.10, 0.35

### Calculating \( E(X) \) and \( E(X^2) \):
- \( E(X) = (20 \times 0.25) + (30 \times 0.10) + (40 \times 0.10) + (50 \times 0.10) + (60 \times 0.10) + (70 \times 0.35) \)
- \( E(X^2) = (20^2 \times 0.25) + (30^2 \times 0.10) + (40^2 \times 0.10) + (50^2 \times 0.10) + (60^2 \times 0.10) + (70^2 \times 0.35) \)

Let's compute these values.The expectation \(E(X)\) and the variance \(\text{Var}(X)\) of the random variable \(X\) are as follows:

- **Expectation (E(X))**: \(47.5\)
- **Variance (Var(X))**: \(418.75\)

Would you like more details or have any questions about this computation?

Here are some related questions to consider:
1. How would you interpret the expectation \(E(X)\) in a real-world scenario?
2. What does the variance \(\text{Var}(X)\) tell us about the distribution of \(X\)?
3. Can you explain how the standard deviation relates to the variance computed here?
4. What would be the impact on \(E(X)\) and \(\text{Var}(X)\) if the probabilities were altered slightly?
5. How would adding another value to the distribution affect \(E(X)\) and \(\text{Var}(X)\)?
6. Could you compute the median of the distribution based on the given probabilities?
7. What is the probability that \(X\) exceeds 50?
8. What other statistical measures could be useful for this data set?

**Tip**: When working with probability distributions, ensure that the probabilities sum up to 1 to maintain the validity of the distribution.

Learn how to calculate the expectation (E(X)) and variance (Var(X)) of a random variable with a given probability distribution. This problem involves applying probability theory concepts, including the expectation and variance formulas. Suitable for advanced high school or college students.

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