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.