https://r2.erweima.ai/mathbot/image/1723524141623s86inq4g.jpg

The image you provided includes a problem involving two random variables, \( X \) and \( Y \), with a joint distribution table. The task is to find \( P(Y = 1) \), the probability that \( Y \) takes the value 1.

### Step-by-Step Solution

To solve this problem, we need to sum the joint probabilities where \( Y = 1 \).

Given the table:

| \(X \backslash Y\) | 1     | 2     | 3     | Marginal \(P(X)\) |
|--------------------|-------|-------|-------|-------------------|
| **1**              | \( \frac{1}{12} \) | \( \frac{1}{12} \) | \( \frac{2}{12} \) | \( \frac{4}{12} \) |
| **2**              | \( \frac{3}{12} \) | \( \frac{1}{12} \) | \( \frac{1}{12} \) | \( \frac{5}{12} \) |
| **3**              | \( \frac{2}{12} \) | \( \frac{0}{12} \) | \( \frac{1}{12} \) | \( \frac{3}{12} \) |
| **Marginal \(P(Y)\)** | \( P(Y = 1) \) | \( P(Y = 2) \) | \( P(Y = 3) \) | \( P(X) \)  |

To find \( P(Y = 1) \), sum the probabilities where \( Y = 1 \) across all possible values of \( X \):

\[
P(Y = 1) = \frac{1}{12} + \frac{3}{12} + \frac{2}{12} = \frac{6}{12} = \frac{1}{2}
\]

So, \( P(Y = 1) = \frac{1}{2} \).

### Would you like more details or have any questions?

Here are some related questions to further explore this topic:
1. How do you find the marginal distribution of \( X \)?
2. What is the joint probability of \( X = 2 \) and \( Y = 2 \)?
3. How do you calculate \( P(X = 1 \text{ and } Y = 3) \)?
4. What is the probability \( P(Y = 2) \)?
5. How do you find the conditional probability \( P(X = 1 | Y = 1) \)?
6. What is the expected value of \( X \) given this joint distribution?
7. How do you determine the independence of \( X \) and \( Y \) using this table?
8. What is the variance of \( Y \) based on the given data?

**Tip:** In joint probability tables, remember that the marginal distributions (sums along rows and columns) can help you quickly find specific probabilities.

Learn how to calculate the probability P(Y = 1) from a joint distribution table involving random variables X and Y. This problem involves finding the marginal probability of Y taking the value 1, with detailed step-by-step solutions and example calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation