To calculate the expected values $E[X]$ and $E[Y]$, we will use the formula for expected value:

$E[X] = \sum x \cdot p(x)$ $E[Y] = \sum y \cdot p(y)$

Calculating $E[X]$:

From the given values of $x$ and $p(x)$:

• $x = 0, p(x) = 0.1$
• $x = 1, p(x) = 0.3$
• $x = 2, p(x) = 0.6$

Now, using the formula for expected value:

$E[X] = (0 \cdot 0.1) + (1 \cdot 0.3) + (2 \cdot 0.6)$ $E[X] = 0 + 0.3 + 1.2 = 1.5$

Calculating $E[Y]$:

From the given values of $y$ and $p(y)$:

• $y = 8, p(y) = 0.4$
• $y = 2, p(y) = 0.1$
• $y = 6, p(y) = 0.2$
• $y = 10, p(y) = 0.3$

Now, using the formula for expected value:

$E[Y] = (8 \cdot 0.4) + (2 \cdot 0.1) + (6 \cdot 0.2) + (10 \cdot 0.3)$ $E[Y] = 3.2 + 0.2 + 1.2 + 3 = 7.6$

Final Answer:

• $E[X] = 1.5$
• $E[Y] = 7.6$

Would you like to explore more about these calculations, or do you have any specific questions about expected values? Here are some related questions to consider:

1. How does the expected value relate to the variance of a random variable?
2. What would happen to $E[X]$ if the probability distribution for $X$ changed?
3. How do you calculate the expected value of the sum of two random variables?
4. What is the difference between discrete and continuous random variables when calculating expected value?
5. Can you compute the covariance between $X$ and $Y$ from their distributions?

Tip: To better understand expected values, think of it as the "weighted average" of all possible outcomes of a random variable, where the probabilities are the weights.