The exercise you’ve shared involves probability distributions, expected values, and variances. Here’s a breakdown of each question:

1. Question 1: The random variable $X$ has a probability distribution given by specific values for $P(X = x)$. You need to calculate the expected value $E(X)$ and the variance $\text{Var}(X)$ for this distribution.

   To solve for $E(X)$:

   • Multiply each possible value of $X$ by its corresponding probability $P(X = x)$.
   • Sum these products to find the expected value.

   For $\text{Var}(X)$:

   • First, calculate $E(X^2)$ by squaring each possible $X$ value, multiplying by $P(X = x)$, and summing these values.
   • Then use the formula $\text{Var}(X) = E(X^2) - (E(X))^2$.

2. Question 2: This is similar to Question 1 but with a different distribution. Follow the same steps to compute $E(X)$ and $\text{Var}(X)$.

3. Question 3: Here, $Y$ is defined as the score when a single, unbiased eight-sided die is rolled. You need to calculate $E(Y)$ and $\text{Var}(Y)$.

   For a fair eight-sided die:

   • Each side (1 through 8) has a probability of $\frac{1}{8}$.
   • The expected value $E(Y) = \frac{1 + 2 + \dots + 8}{8}$.
   • To find $\text{Var}(Y)$, use the formula for the variance of a uniform distribution or compute $E(Y^2) - (E(Y))^2$.

Would you like a detailed step-by-step solution for each part?

Here are 5 questions to deepen understanding:

1. How is the expected value affected by the probabilities associated with each outcome?
2. What is the significance of variance in a probability distribution?
3. How does calculating $E(X^2)$ differ from calculating $(E(X))^2$?
4. Why is the variance for a uniform distribution calculated differently from other distributions?
5. How would changing one probability in the distribution affect $E(X)$ and $\text{Var}(X)$?

Tip: When dealing with probability distributions, always ensure that the sum of all probabilities is 1 before calculating expected values and variances.