Let X, and Y be two random variables with respective means x, and Hy, and standard deviations ox, and oy. We define two new RVs as follows: V = X +aY, and W = XaY, where a is a real constant. Find the value of a for which V and W are uncorrelated.
Let X, and Y be two random variables with respective means x, and Hy, and standard deviations ox, and oy. We define two new RVs as follows: V = X +aY, and W = XaY, where a is a real constant. Find the value of a for which V and W are uncorrelated.
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![**Title: Understanding Uncorrelated Random Variables through an Example**
---
**Introduction to Random Variables:**
Let \( X \) and \( Y \) be two random variables with respective means \( \mu_X \) and \( \mu_Y \), and standard deviations \( \sigma_X \) and \( \sigma_Y \).
**Defining New Random Variables:**
We define two new random variables (RVs) as follows:
- \( V = X + aY \)
- \( W = X - aY \)
Here, \( a \) is a real constant.
**Objective:**
Determine the value of \( a \) for which \( V \) and \( W \) are uncorrelated.
---
**Detailed Explanation:**
To find the value of \( a \) for which \( V \) and \( W \) are uncorrelated, we need to ensure that their covariance, \( \text{Cov}(V, W) \), is zero.
Mathematically, calculating the covariance between \( V \) and \( W \):
\[
\text{Cov}(V, W) = \text{Cov}(X + aY, X - aY)
\]
We apply the bilinearity property of covariance:
\[
\text{Cov}(V, W) = \text{Cov}(X, X) - a \text{Cov}(X, Y) + a \text{Cov}(Y, X) - a^2 \text{Cov}(Y, Y)
\]
Using the property that \( \text{Cov}(X, X) = \text{Var}(X) = \sigma_X^2 \), \( \text{Cov}(Y, Y) = \text{Var}(Y) = \sigma_Y^2 \), and since covariance is symmetric \( \text{Cov}(X, Y) = \text{Cov}(Y, X) \):
\[
\text{Cov}(V, W) = \sigma_X^2 - a \sigma_{XY} + a \sigma_{XY} - a^2 \sigma_Y^2
\]
Simplifying this expression:
\[
\text{Cov}(V, W) = \sigma_X^2 - a^2 \sigma_Y^2
\]
Setting this equal to zero for uncorrelated variables:
\[
\sigma_X^2 - a^2 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2F8a0acb63-84d7-428a-95a0-1b2466199014%2Fkgoyny9_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Uncorrelated Random Variables through an Example**
---
**Introduction to Random Variables:**
Let \( X \) and \( Y \) be two random variables with respective means \( \mu_X \) and \( \mu_Y \), and standard deviations \( \sigma_X \) and \( \sigma_Y \).
**Defining New Random Variables:**
We define two new random variables (RVs) as follows:
- \( V = X + aY \)
- \( W = X - aY \)
Here, \( a \) is a real constant.
**Objective:**
Determine the value of \( a \) for which \( V \) and \( W \) are uncorrelated.
---
**Detailed Explanation:**
To find the value of \( a \) for which \( V \) and \( W \) are uncorrelated, we need to ensure that their covariance, \( \text{Cov}(V, W) \), is zero.
Mathematically, calculating the covariance between \( V \) and \( W \):
\[
\text{Cov}(V, W) = \text{Cov}(X + aY, X - aY)
\]
We apply the bilinearity property of covariance:
\[
\text{Cov}(V, W) = \text{Cov}(X, X) - a \text{Cov}(X, Y) + a \text{Cov}(Y, X) - a^2 \text{Cov}(Y, Y)
\]
Using the property that \( \text{Cov}(X, X) = \text{Var}(X) = \sigma_X^2 \), \( \text{Cov}(Y, Y) = \text{Var}(Y) = \sigma_Y^2 \), and since covariance is symmetric \( \text{Cov}(X, Y) = \text{Cov}(Y, X) \):
\[
\text{Cov}(V, W) = \sigma_X^2 - a \sigma_{XY} + a \sigma_{XY} - a^2 \sigma_Y^2
\]
Simplifying this expression:
\[
\text{Cov}(V, W) = \sigma_X^2 - a^2 \sigma_Y^2
\]
Setting this equal to zero for uncorrelated variables:
\[
\sigma_X^2 - a^2 \
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