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.