10.	If X,Y,Z are uncorrelated random variables with S.D 5, 12 and 9 respectively and if
U = X+Y and V= Y+Z , then evaluate the Correlation coefficient between U and V .

Given that \( X \), \( Y \), and \( Z \) are uncorrelated random variables with standard deviations \( \sigma_X = 5 \), \( \sigma_Y = 12 \), and \( \sigma_Z = 9 \), respectively, and that \( U = X + Y \) and \( V = Y + Z \), we need to evaluate the correlation coefficient between \( U \) and \( V \).

### Step 1: Calculate the variance of \( U \) and \( V \).
The variance of a sum of uncorrelated random variables is the sum of their variances.

\[
\text{Var}(U) = \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)
\]

\[
\text{Var}(V) = \text{Var}(Y + Z) = \text{Var}(Y) + \text{Var}(Z)
\]

Given the standard deviations, the variances are:

\[
\text{Var}(X) = \sigma_X^2 = 5^2 = 25
\]

\[
\text{Var}(Y) = \sigma_Y^2 = 12^2 = 144
\]

\[
\text{Var}(Z) = \sigma_Z^2 = 9^2 = 81
\]

Thus,

\[
\text{Var}(U) = 25 + 144 = 169
\]

\[
\text{Var}(V) = 144 + 81 = 225
\]

### Step 2: Calculate the covariance \( \text{Cov}(U, V) \).
Since \( X \), \( Y \), and \( Z \) are uncorrelated, the covariance \( \text{Cov}(U, V) \) is:

\[
\text{Cov}(U, V) = \text{Cov}(X+Y, Y+Z) = \text{Cov}(X, Y) + \text{Cov}(X, Z) + \text{Cov}(Y, Y) + \text{Cov}(Y, Z)
\]

Given that \( X \) is uncorrelated with \( Y \) and \( Z \), and \( Y \) is uncorrelated with \( Z \), we have:

\[
\text{Cov}(U, V) = 0 + 0 + \text{Var}(Y) + 0 = \text{Var}(Y) = 144
\]

### Step 3: Calculate the correlation coefficient.
The correlation coefficient \( \rho(U, V) \) is given by:

\[
\rho(U, V) = \frac{\text{Cov}(U, V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}}
\]

Substituting the values we found:

\[
\rho(U, V) = \frac{144}{\sqrt{169 \times 225}}
\]

\[
\rho(U, V) = \frac{144}{\sqrt{38025}} = \frac{144}{195} \approx 0.738
\]

Thus, the correlation coefficient between \( U \) and \( V \) is approximately \( 0.738 \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the impact of correlation between two variables on their covariance?
2. How would the correlation coefficient change if \( X \) and \( Z \) were correlated?
3. What if \( U \) and \( V \) were defined differently, such as \( U = X + 2Y \)?
4. How does the standard deviation of a sum of uncorrelated variables differ from that of correlated variables?
5. What real-world applications involve calculating the correlation coefficient between two sums of random variables?

**Tip:** Understanding the relationship between covariance and correlation is crucial for interpreting how variables interact in statistical models.

Learn how to evaluate the correlation coefficient between sums of uncorrelated random variables X, Y, and Z, each with specified standard deviations. Detailed step-by-step solution includes calculating variances, covariance, and correlation coefficient. Suitable for advanced high school or college students studying statistics and probability.

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