IDNLearn.com provides a collaborative environment for finding accurate answers. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
To solve the given question, let's address each part step by step.
Part (a): Calculating \( p(u, v) \)
We need to find the joint probability \( P(U=u, V=v) \) for the number of red \( (U) \) and green \( (V) \) ribbons selected.
Given:
- Total number of ribbons = 15
- Number of blue ribbons = 9
- Number of green ribbons = 4
- Number of red ribbons = 15 - 9 - 4 = 2
- Number of ribbons selected = 5
Using combinatorics, we can calculate the probability of selecting \( u \) red ribbons and \( v \) green ribbons, while the remaining ribbons \( (5 - u - v) \) are blue.
Let's calculate an example probability \( P(U=1, V=2) \).
[tex]\[ p(u, v) \][/tex]
- Number of ways to choose 1 red ribbon from 2 red ribbons: \(\binom{2}{1}\)
- Number of ways to choose 2 green ribbons from 4 green ribbons: \(\binom{4}{2}\)
- Number of ways to choose 2 blue ribbons from 9 blue ribbons: \(\binom{9}{2}\)
- Total number of ways to choose 5 ribbons from 15 ribbons: \(\binom{15}{5}\)
Using these combinations, we calculate:
[tex]\[ p(1, 2) = \frac{\binom{2}{1} \cdot \binom{4}{2} \cdot \binom{9}{2}}{\binom{15}{5}} \][/tex]
The result for this example calculation is:
[tex]\[ p(1, 2) \approx 0.143856 \][/tex]
Part (b): Calculating the Correlation Between \( U \) and \( V \), \( \text{Corr}(U, V) \)
To find the correlation, we need to compute the following expectations and variances:
- \( E(U) \): Expected value of \( U \)
- \( E(V) \): Expected value of \( V \)
- \( E(UV) \): Expected value of the product \( UV \)
- \( \text{Var}(U) \): Variance of \( U \)
- \( \text{Var}(V) \): Variance of \( V \)
Then, the correlation \( \text{Corr}(U, V) \) is given by:
[tex]\[ \text{Corr}(U, V) = \frac{E(UV) - E(U)E(V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}} \][/tex]
From our detailed calculations, we have the following values:
- \( E(U) \approx 0.6667 \)
- \( E(V) \approx 1.3333 \)
- \( E(UV) \approx 0.7619 \)
- \( \text{Var}(U) \approx 0.4127 \)
- \( \text{Var}(V) \approx 0.6984 \)
Thus, the correlation is:
[tex]\[ \text{Corr}(U, V) \approx \frac{0.7619 - (0.6667 \times 1.3333)}{\sqrt{0.4127 \times 0.6984}} \][/tex]
After evaluating the above expression, the correlation coefficient is found to be approximately:
[tex]\[ \text{Corr}(U, V) \approx -0.2365 \][/tex]
So, summarizing the results:
- The joint probability \( p(U=1, V=2) \approx 0.1439 \)
- The correlation [tex]\( \text{Corr}(U, V) \approx -0.2365 \)[/tex]
Part (a): Calculating \( p(u, v) \)
We need to find the joint probability \( P(U=u, V=v) \) for the number of red \( (U) \) and green \( (V) \) ribbons selected.
Given:
- Total number of ribbons = 15
- Number of blue ribbons = 9
- Number of green ribbons = 4
- Number of red ribbons = 15 - 9 - 4 = 2
- Number of ribbons selected = 5
Using combinatorics, we can calculate the probability of selecting \( u \) red ribbons and \( v \) green ribbons, while the remaining ribbons \( (5 - u - v) \) are blue.
Let's calculate an example probability \( P(U=1, V=2) \).
[tex]\[ p(u, v) \][/tex]
- Number of ways to choose 1 red ribbon from 2 red ribbons: \(\binom{2}{1}\)
- Number of ways to choose 2 green ribbons from 4 green ribbons: \(\binom{4}{2}\)
- Number of ways to choose 2 blue ribbons from 9 blue ribbons: \(\binom{9}{2}\)
- Total number of ways to choose 5 ribbons from 15 ribbons: \(\binom{15}{5}\)
Using these combinations, we calculate:
[tex]\[ p(1, 2) = \frac{\binom{2}{1} \cdot \binom{4}{2} \cdot \binom{9}{2}}{\binom{15}{5}} \][/tex]
The result for this example calculation is:
[tex]\[ p(1, 2) \approx 0.143856 \][/tex]
Part (b): Calculating the Correlation Between \( U \) and \( V \), \( \text{Corr}(U, V) \)
To find the correlation, we need to compute the following expectations and variances:
- \( E(U) \): Expected value of \( U \)
- \( E(V) \): Expected value of \( V \)
- \( E(UV) \): Expected value of the product \( UV \)
- \( \text{Var}(U) \): Variance of \( U \)
- \( \text{Var}(V) \): Variance of \( V \)
Then, the correlation \( \text{Corr}(U, V) \) is given by:
[tex]\[ \text{Corr}(U, V) = \frac{E(UV) - E(U)E(V)}{\sqrt{\text{Var}(U) \cdot \text{Var}(V)}} \][/tex]
From our detailed calculations, we have the following values:
- \( E(U) \approx 0.6667 \)
- \( E(V) \approx 1.3333 \)
- \( E(UV) \approx 0.7619 \)
- \( \text{Var}(U) \approx 0.4127 \)
- \( \text{Var}(V) \approx 0.6984 \)
Thus, the correlation is:
[tex]\[ \text{Corr}(U, V) \approx \frac{0.7619 - (0.6667 \times 1.3333)}{\sqrt{0.4127 \times 0.6984}} \][/tex]
After evaluating the above expression, the correlation coefficient is found to be approximately:
[tex]\[ \text{Corr}(U, V) \approx -0.2365 \][/tex]
So, summarizing the results:
- The joint probability \( p(U=1, V=2) \approx 0.1439 \)
- The correlation [tex]\( \text{Corr}(U, V) \approx -0.2365 \)[/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.
