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The problem in the image involves two discrete random variables \( X \) and \( Y \) with a joint probability density function \( f_{X,Y}(x,y) \), given by:

\[
f_{X,Y}(x,y) = 0.4 \delta(x + \alpha)\delta(y - 2) + 0.3 \delta(x - \alpha)\delta(y - 2) + 0.1 \delta(x - \alpha)\delta(y - \alpha) + 0.2 \delta(x - 1)\delta(y - 1)
\]

Where \( \delta \) represents the Dirac delta function, and you are asked to:
1. Determine the value of \( \alpha \) that minimizes the correlation between \( X \) and \( Y \).
2. Find the minimum correlation.
3. Answer whether \( X \) and \( Y \) are orthogonal.

### Solution Plan:

1. **Compute the marginal distributions** of \( X \) and \( Y \) by summing the joint density over the other variable.
2. **Calculate the mean and variance** of \( X \) and \( Y \) based on the marginal distributions.
3. **Find the covariance** between \( X \) and \( Y \).
4. **Determine the correlation coefficient** \( \rho(X, Y) \).
5. **Minimize the correlation** by finding the value of \( \alpha \) that gives the minimum correlation.
6. **Check orthogonality**: \( X \) and \( Y \) are orthogonal if the covariance \( \text{Cov}(X,Y) = 0 \).

Let's begin solving this step by step. 

### Step 1: Compute the Marginal Distributions

The joint distribution is composed of Dirac delta functions, so each term represents a point mass at a specific location. The marginal distributions for \( X \) and \( Y \) can be computed by integrating the joint density over one variable.

- Marginal of \( X \):
\[
f_X(x) = \sum_y f_{X,Y}(x,y)
\]

- Marginal of \( Y \):
\[
f_Y(y) = \sum_x f_{X,Y}(x,y)
\]

Let me first calculate the marginal distributions.The marginal distributions of \( X \) and \( Y \) are as follows:

- **Marginal distribution of \( X \):**
\[
f_X(x) = 0.4 \delta(x + \alpha) + 0.3 \delta(x - \alpha) + 0.1 \delta(x - \alpha) + 0.2 \delta(x - 1)
\]
This can be simplified to:
\[
f_X(x) = 0.4 \delta(x + \alpha) + 0.4 \delta(x - \alpha) + 0.2 \delta(x - 1)
\]

- **Marginal distribution of \( Y \):**
\[
f_Y(y) = 0.7 \delta(y - 2) + 0.1 \delta(y - \alpha) + 0.2 \delta(y - 1)
\]

Now, with these marginals, I can proceed to compute the expectations (mean), variances, and covariance to determine the correlation between \( X \) and \( Y \). This will help in finding the value of \( \alpha \) that minimizes the correlation. Let’s compute the expectations and covariance next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain any of the steps or reasoning I outlined earlier, or assist with another question related to this problem?

Here are some questions to expand on this topic:
1. What is the definition of correlation and how is it computed?
2. How do we interpret Dirac delta functions in probability distributions?
3. What does orthogonality between two random variables imply?
4. How does the value of \( \alpha \) affect the correlation between \( X \) and \( Y \)?
5. How can covariance help in determining the dependence between two variables?

**Tip:** Minimizing the correlation between two variables doesn't necessarily make them independent, but it reduces their linear dependence.

Discrete random variables X and Y have the joint density f_X,Y(x,y) = 0.4 δ(x+α) δ(y−2) + 0.3 δ(x−α) δ(y−2) + 0.1 δ(x−α) δ(y−α) + 0.2 δ(x−1) δ(y−1). Determine the value of α, if any, that minimizes the correlation between X and Y. And find the minimum correlation. Are X and Y orthogonal?

This problem explores how to minimize the correlation between discrete random variables X and Y with a given joint density function using Dirac delta functions. Learn how to compute marginal distributions, expectations, and determine if the variables are orthogonal.

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