X is a random variable with mean 5 and standard deviation 2. Find i. E(Y) and Var(Y) for: a) Y=2x+3 b)Y=2x-3 c)Y=(x-5)?2

To solve for \( E(Y) \) and \( \text{Var}(Y) \) in each case, we will use the properties of expected values and variances.

### Given:
- \( X \) is a random variable with mean \( E(X) = 5 \) and standard deviation \( \sigma_X = 2 \).
- The variance of \( X \) is \( \text{Var}(X) = \sigma_X^2 = 4 \).

### (a) For \( Y = 2X + 3 \):
1. **Expected Value \( E(Y) \)**:
   \[
   E(Y) = E(2X + 3) = 2 \cdot E(X) + 3 = 2 \cdot 5 + 3 = 10 + 3 = 13
   \]

2. **Variance \( \text{Var}(Y) \)**:
   \[
   \text{Var}(Y) = \text{Var}(2X + 3) = 2^2 \cdot \text{Var}(X) = 4 \cdot 4 = 16
   \]

   **Result for (a):**
   \[
   E(Y) = 13, \quad \text{Var}(Y) = 16
   \]

### (b) For \( Y = 2X - 3 \):
1. **Expected Value \( E(Y) \)**:
   \[
   E(Y) = E(2X - 3) = 2 \cdot E(X) - 3 = 2 \cdot 5 - 3 = 10 - 3 = 7
   \]

2. **Variance \( \text{Var}(Y) \)**:
   \[
   \text{Var}(Y) = \text{Var}(2X - 3) = 2^2 \cdot \text{Var}(X) = 4 \cdot 4 = 16
   \]

   **Result for (b):**
   \[
   E(Y) = 7, \quad \text{Var}(Y) = 16
   \]

### (c) For \( Y = (X - 5)^2 \):
1. **Expected Value \( E(Y) \)**:
   - Here, \( Y = (X - 5)^2 \) represents the square of a centered random variable.
   - Note that \( X - 5 \) has a mean of \( 0 \) and variance equal to \( \text{Var}(X) = 4 \).
   - The expected value of \( Y \) is the variance of \( X \):
     \[
     E(Y) = \text{Var}(X) = 4
     \]

2. **Variance \( \text{Var}(Y) \)**:
   - To find \( \text{Var}(Y) \), we would need additional information about the distribution of \( X \), as calculating \( \text{Var}((X-5)^2) \) involves higher moments of \( X \).
   - Without additional information on the distribution of \( X \), we cannot determine \( \text{Var}(Y) \) for this case.

   **Result for (c):**
   \[
   E(Y) = 4, \quad \text{Var}(Y) = \text{(Cannot be determined with given information)}
   \]

---

Would you like more details on these calculations, or do you have any questions?

### Related Questions
1. How would the values change if \( X \) had a different standard deviation?
2. How does the linear transformation affect the variance in cases (a) and (b)?
3. Why does the variance formula for \( Y = 2X + 3 \) involve squaring the coefficient of \( X \)?
4. What assumptions are needed to calculate higher moments of \( Y = (X-5)^2 \)?
5. How would you find \( \text{Var}(Y) \) if \( X \) were normally distributed?

### Tip
For linear transformations \( Y = aX + b \), remember that \( E(Y) = aE(X) + b \) and \( \text{Var}(Y) = a^2 \text{Var}(X) \).

X is a random variable with mean 5 and standard deviation 2. Find i. E(Y) and Var(Y) for: a) Y=2x+3 b) Y=2x-3 c) Y=(x-5)^2

Learn how to calculate the expected value (E(Y)) and variance (Var(Y)) for functions of a random variable X with mean 5 and standard deviation 2. This tutorial explores linear transformations and moment calculations, with a detailed solution for each case: Y = 2X + 3, Y = 2X - 3, and Y = (X - 5)^2. Ideal for college-level 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