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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]
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