https://file.aiquickdraw.com/mathbot/file/1726072955535cawu5raj.jpg

We are tasked with finding the expected value of \( Z \), given that \( Y = 10 \), based on the joint probability distribution.

Let's analyze the provided table first. We know that:

- \( Y \) takes values 10 and 20.
- \( Z \) takes values 29 and 21.

The relevant row for \( Y = 10 \) has the probabilities:

- \( P(Y = 10, Z = 29) = 0.2 \)
- \( P(Y = 10, Z = 21) = 0.1 \)

### Step 1: Normalize the conditional probabilities
We first need to compute \( P(Z = z \mid Y = 10) \), which is the probability of \( Z \) given that \( Y = 10 \).

\[
P(Z = 29 \mid Y = 10) = \frac{P(Y = 10, Z = 29)}{P(Y = 10)} = \frac{0.2}{0.2 + 0.1} = \frac{0.2}{0.3} = \frac{2}{3}
\]
\[
P(Z = 21 \mid Y = 10) = \frac{P(Y = 10, Z = 21)}{P(Y = 10)} = \frac{0.1}{0.3} = \frac{1}{3}
\]

### Step 2: Compute the expected value
The expected value of \( Z \), given that \( Y = 10 \), is calculated as:

\[
E[Z \mid Y = 10] = P(Z = 29 \mid Y = 10) \times 29 + P(Z = 21 \mid Y = 10) \times 21
\]
Substituting the values:

\[
E[Z \mid Y = 10] = \frac{2}{3} \times 29 + \frac{1}{3} \times 21
\]

Now, let's calculate this.The expected value of \( Z \), given that \( Y = 10 \), is approximately 26.33.

Would you like to delve deeper into any of the steps? Here are five related questions that can expand your understanding:

1. How would you calculate the marginal probability \( P(Z = 29) \)?
2. What is the expected value of \( Z \) given \( Y = 20 \)?
3. How do you calculate conditional probability in general?
4. Can the expected value change if the joint probabilities for \( Z \) are different?
5. What is the significance of the joint probability distribution in statistics?

**Tip:** Always ensure to normalize probabilities when working with conditional expectations to get accurate results.

What is the expected value of Z, given that Y = 10, based on the provided joint probability distribution table?

Learn how to calculate the expected value of Z, given that Y = 10, using a joint probability distribution table. This problem explores key concepts like conditional probability and expected value. Suitable for advanced high school or introductory college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation