https://file.aiquickdraw.com/mathbot/file/1731048377407fz8x7shc.jpg

The exercise you’ve shared involves probability distributions, expected values, and variances. Here’s a breakdown of each question:

1. **Question 1**: The random variable \(X\) has a probability distribution given by specific values for \( P(X = x) \). You need to calculate the expected value \(E(X)\) and the variance \( \text{Var}(X) \) for this distribution.

   To solve for \( E(X) \):
   - Multiply each possible value of \(X\) by its corresponding probability \(P(X = x)\).
   - Sum these products to find the expected value.

   For \( \text{Var}(X) \):
   - First, calculate \(E(X^2)\) by squaring each possible \(X\) value, multiplying by \(P(X = x)\), and summing these values.
   - Then use the formula \( \text{Var}(X) = E(X^2) - (E(X))^2 \).

2. **Question 2**: This is similar to Question 1 but with a different distribution. Follow the same steps to compute \(E(X)\) and \( \text{Var}(X) \).

3. **Question 3**: Here, \(Y\) is defined as the score when a single, unbiased eight-sided die is rolled. You need to calculate \(E(Y)\) and \( \text{Var}(Y) \).
   
   For a fair eight-sided die:
   - Each side (1 through 8) has a probability of \( \frac{1}{8} \).
   - The expected value \(E(Y) = \frac{1 + 2 + \dots + 8}{8}\).
   - To find \( \text{Var}(Y) \), use the formula for the variance of a uniform distribution or compute \(E(Y^2) - (E(Y))^2\).

Would you like a detailed step-by-step solution for each part? 

Here are 5 questions to deepen understanding:

1. How is the expected value affected by the probabilities associated with each outcome?
2. What is the significance of variance in a probability distribution?
3. How does calculating \( E(X^2) \) differ from calculating \( (E(X))^2 \)?
4. Why is the variance for a uniform distribution calculated differently from other distributions?
5. How would changing one probability in the distribution affect \(E(X)\) and \( \text{Var}(X) \)?

**Tip**: When dealing with probability distributions, always ensure that the sum of all probabilities is 1 before calculating expected values and variances.

Find the expected value (E(X)) and variance (Var(X)) for given probability distributions, and calculate E(Y) and Var(Y) for the score of a single, unbiased eight-sided die roll.

Understand how to calculate the expected value and variance for probability distributions, including examples with a random variable and an unbiased eight-sided die. Ideal for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation