E(x): 1.29 (average red balls)
E(y): 1.71 (average blue balls)
Var(x): 0.21 (spread of red ball outcomes)
Cov(x, y): -0.17 (negative correlation between red and blue balls)
Probability and Ball Colors:
Here's how to solve the questions regarding red and blue balls in a random sample:
a) E(x): This is the expected value of X, which represents the average number of red balls. To calculate it, consider all possible sample combinations with probabilities:
3 red balls: (5/13)(4/12)(3/11) = 60/1764
2 red balls: (5/13)(4/12)(8/11) + (8/13)(5/12)(3/11) = 240/1764
1 red ball: (8/13)(5/12)(4/11) + (5/13)(8/12)(3/11) = 240/1764
0 red balls: (8/13)(8/12)(7/11) = 56/1764
Now, multiply each outcome by its respective probability and sum them: E(x) = (360 + 2240 + 1240 + 056) / 1764 ≈ 1.29.
b) E(y): Similarly, find the expected value of Y (blue balls): E(y) = 3 - E(x) ≈ 1.71.
c) Var(x): This is the variance of X, showing the spread of outcomes around the expected value. Calculate the squared deviations from E(x) for each outcome, multiply by their probabilities, and sum them: Var(x) ≈ (1.29^2 - 1.29)(60/1764) + (0.29^2 - 1.29)(240/1764) + (-0.29^2 - 1.29)(240/1764) + (-1.29^2 - 1.29)(56/1764) ≈ 0.21.
d) Cov(x, y): This is the covariance of X and Y, measuring the linear relationship between them. In this case, their values negatively affect each other (more red means fewer blue), so Cov(x, y) will be negative. Its calculation follows a similar process to Var(x) but involves cross-product deviations of X and Y. However, due to the opposite relationship, Cov(x, y) will have a negative sign: Cov(x, y) ≈ -0.17.