1. The probability distribution table for X and Y is presented below. Y\X 0 1 2 0 0.05 0.1 0.03 1 0.21 0.11 0.19 2 0.08 0.15 0.08 1-a) compute COV(X, Y) 1-b) compute COV(X 2 , Y)



my answers for 
a) 0.0252 b) 0.1272




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.

1. The probability distribution table for X and Y is presented below. Y\X 0 1 2 0 0.05 0.1 0.03 1 0.21 0.11 0.19 2 0.08 0.15 0.08 1-a) compute COV(X, Y) 1-b) compute COV(X^2, Y). My answers: a) 0.0252 b) 0.1272

Explore how to compute Cov(X, Y) and Cov(X², Y) using a given probability distribution table. Learn step-by-step calculations involving expected values and covariance formulas. Suitable for undergraduate students studying statistics or probability.

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