The joint probability distribution table for two random variables X and Y is as follows
Y
X -2 -1 4 5
1 0.1 0.2 0 0.3
0.1 0.1 0
2 0.2 Determine the marginal probability distributions of X and Y. Also compute (a) Expectations
of X and Y (b) Standard deviation of X and Y (c) Covariance of X and Y (d) Correlation of X
and Y.

We are given a joint probability distribution table for two random variables \( X \) and \( Y \). Let's extract the values first to determine the marginal distributions, expectations, standard deviations, covariance, and correlation.

### 1. **Joint Probability Distribution Table**

| **Y / X** | -2    | -1   | 4     | 5    | Marginal \( P_X \) |
|:---------:|-------|------|-------|------|---------------------|
| **1**     | 0.1   | 0.2  | 0     | 0.3  | \( P(X = 1) = 0.6 \) |
| **0**     | 0.1   | 0.1  | 0     | 0    | \( P(X = 0) = 0.2 \) |
| **2**     | 0     | 0    | 0.2   | 0    | \( P(X = 2) = 0.2 \) |

#### Step 1: **Marginal Probability Distributions of X and Y**

- **Marginal probability distribution of \( X \)**
  
We sum over all the values of \( Y \) for each value of \( X \):

\[
P(X = 1) = 0.1 + 0.2 + 0 + 0.3 = 0.6
\]
\[
P(X = 0) = 0.1 + 0.1 + 0 + 0 = 0.2
\]
\[
P(X = 2) = 0 + 0 + 0.2 + 0 = 0.2
\]

Thus, the marginal distribution of \( X \) is:

| \( X \)  | 1   | 0   | 2   |
|:--------:|-----|-----|-----|
| \( P_X \)| 0.6 | 0.2 | 0.2 |

- **Marginal probability distribution of \( Y \)**

We sum over all the values of \( X \) for each value of \( Y \):

\[
P(Y = -2) = 0.1 + 0.1 + 0 = 0.2
\]
\[
P(Y = -1) = 0.2 + 0.1 + 0 = 0.3
\]
\[
P(Y = 4) = 0 + 0 + 0.2 = 0.2
\]
\[
P(Y = 5) = 0.3 + 0 + 0 = 0.3
\]

Thus, the marginal distribution of \( Y \) is:

| \( Y \)  | -2  | -1  | 4   | 5   |
|:--------:|-----|-----|-----|-----|
| \( P_Y \)| 0.2 | 0.3 | 0.2 | 0.3 |

#### Step 2: **Expectations of \( X \) and \( Y \)**

- **Expectation of \( X \)**

\[
E(X) = \sum_{x} x \cdot P_X(x) = 1 \cdot 0.6 + 0 \cdot 0.2 + 2 \cdot 0.2 = 0.6 + 0.4 = 1
\]

- **Expectation of \( Y \)**

\[
E(Y) = \sum_{y} y \cdot P_Y(y) = (-2) \cdot 0.2 + (-1) \cdot 0.3 + 4 \cdot 0.2 + 5 \cdot 0.3
\]
\[
E(Y) = -0.4 - 0.3 + 0.8 + 1.5 = 1.6
\]

#### Step 3: **Standard Deviation of \( X \) and \( Y \)**

- **Variance of \( X \)**

\

The joint probability distribution table for two random variables X and Y is as follows
Y
X -2 -1 4 5
1 0.1 0.2 0 0.3
0.1 0.1 0
2 0.2 Determine the marginal probability distributions of X and Y. Also compute (a) Expectations of X and Y (b) Standard deviation of X and Y (c) Covariance of X and Y (d) Correlation of X and Y.

Explore how to compute the marginal probability distributions of X and Y from a joint probability table. This detailed solution includes the calculation of expectations, standard deviations, covariance, and correlation between two random variables. Perfect for undergraduate students studying probability and statistics.

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