Let's go over your answers for both parts of the problem:

1-a) Compute Cov(X, Y):

The formula for the covariance between two random variables $X$ and $Y$ is:

$\text{Cov}(X, Y) = E(XY) - E(X)E(Y)$

We need the following components to compute this:

1. $E(X)$: The expected value (mean) of $X$.
2. $E(Y)$: The expected value (mean) of $Y$.
3. $E(XY)$: The expected value of the product $XY$.

Step 1: Compute $E(X)$

The marginal probability distribution of $X$ is found by summing over the values of $Y$:

$P(X=0) = 0.05 + 0.1 + 0.03 = 0.18$ $P(X=1) = 0.21 + 0.11 + 0.19 = 0.51$ $P(X=2) = 0.08 + 0.15 + 0.08 = 0.31$

So,

$E(X) = 0 \times 0.18 + 1 \times 0.51 + 2 \times 0.31 = 0.51 + 0.62 = 1.13$

Step 2: Compute $E(Y)$

Similarly, find the marginal probability distribution of $Y$:

$P(Y=0) = 0.05 + 0.21 + 0.08 = 0.34$ $P(Y=1) = 0.1 + 0.11 + 0.15 = 0.36$ $P(Y=2) = 0.03 + 0.19 + 0.08 = 0.30$

So,

$E(Y) = 0 \times 0.34 + 1 \times 0.36 + 2 \times 0.30 = 0.36 + 0.60 = 0.96$

Step 3: Compute $E(XY)$

Now, compute the expected value of the product $XY$:

$E(XY) = \sum_{x,y} x y P(X=x, Y=y)$ $E(XY) = (0 \times 0 \times 0.05) + (0 \times 1 \times 0.1) + (0 \times 2 \times 0.03) + (1 \times 0 \times 0.21) + (1 \times 1 \times 0.11) + (1 \times 2 \times 0.19)$ $+ (2 \times 0 \times 0.08) + (2 \times 1 \times 0.15) + (2 \times 2 \times 0.08)$ $E(XY) = 0 + 0 + 0 + 0 + 0.11 + 0.38 + 0 + 0.30 + 0.32 = 1.11$

Step 4: Compute $\text{Cov}(X, Y)$

Now, apply the covariance formula:

$\text{Cov}(X, Y) = E(XY) - E(X)E(Y)$ $\text{Cov}(X, Y) = 1.11 - (1.13 \times 0.96) = 1.11 - 1.0848 = 0.0252$

So, your answer for part 1-a is correct: Cov(X, Y) = 0.0252.

---

1-b) Compute Cov(X^2, Y):

The covariance formula is still the same:

$\text{Cov}(X^2, Y) = E(X^2 Y) - E(X^2) E(Y)$

We now need to compute $E(X^2)$ and $E(X^2 Y)$.

Step 1: Compute $E(X^2)$

$E(X^2) = \sum_x x^2 P(X=x)$ $E(X^2) = 0^2 \times 0.18 + 1^2 \times 0.51 + 2^2 \times 0.31 = 0 + 0.51 + 1.24 = 1.75$

Step 2: Compute $E(X^2 Y)$

$E(X^2 Y) = \sum_{x,y} x^2 y P(X=x, Y=y)$ $E(X^2 Y) = (0^2 \times 0 \times 0.05) + (0^2 \times 1 \times 0.1) + (0^2 \times 2 \times 0.03) + (1^2 \times 0 \times 0.21)$ $+ (1^2 \times 1 \times 0.11) + (1^2 \times 2 \times 0.19) + (2^2 \times 0 \times 0.08) + (2^2 \times 1 \times 0.15) + (2^2 \times 2 \times 0.08)$ $E(X^2 Y) = 0 + 0 + 0 + 0 + 0.11 + 0.38 + 0 + 0.60 + 0.64 = 1.73$

Step 3: Compute $\text{Cov}(X^2, Y)$

$\text{Cov}(X^2, Y) = E(X^2 Y) - E(X^2)E(Y)$ $\text{Cov}(X^2, Y) = 1.73 - (1.75 \times 0.96) = 1.73 - 1.68 = 0.05$

It seems like your value for 1-b was slightly off. The correct answer for Cov(X², Y) = 0.05.

---

Summary:

• 1-a: Your answer is correct: $\text{Cov}(X, Y) = 0.0252$
• 1-b: The correct answer is: $\text{Cov}(X^2, Y) = 0.05$

Would you like a deeper breakdown of any of these steps or further details?

Related Questions:

1. How does covariance indicate the relationship between two variables?
2. Can the covariance of two independent variables ever be non-zero?
3. How would you interpret a covariance of 0?
4. What is the difference between covariance and correlation?
5. How does the covariance of transformed variables, such as $X^2$, change compared to the original variables?

Tip: Covariance only measures the linear relationship between two variables. Non-linear relationships require more advanced techniques like correlation or regression models.