Final answer:
To find the expected value and variance of X, first calculate the probability of each possible outcome. Then, multiply each probability by the corresponding value of X. The expected value is 1232 and variance is 22,279,840.
Step-by-step explanation:
Expected Value:
To find the expected value of X, we need to calculate the probability of each possible outcome and multiply it by the value of that outcome. In this case, X can take on values from 0 to 4, representing the number of complete pairs of shoes among the selected shoes.
The probability of selecting 0 complete pairs can be calculated as the number of ways to choose 4 shoes without matching pairs divided by the total number of ways to choose 4 shoes. There are 8 pairs of shoes, so there are 16 individual shoes. The number of ways to choose 4 shoes without matching pairs is given by the combination formula C(16, 4) = 1820.
The probability of selecting 1 complete pair can be calculated as the number of ways to choose 1 pair from the 8 pairs, multiplied by the number of ways to choose 2 individual shoes from the remaining 14 shoes.
This is equal to 8 * C(14, 2) = 8 * 91 = 728.
Similarly, the probabilities for selecting 2 complete pairs, 3 complete pairs, and 4 complete pairs can be calculated as 28, 56, and 70 respectively.
To calculate the expected value, we multiply each probability by the corresponding value of X and sum them up. The expected value of X is given by
E(X) = (0 * P(X = 0)) + (1 * P(X = 1)) + (2 * P(X = 2)) + (3 * P(X = 3)) + (4 * P(X = 4)).
Calculating this gives us
E(X) = (0 * 1820) + (1 * 728) + (2 * 28) + (3 * 56) + (4 * 70)
E(X) = 0 + 728 + 56 + 168 + 280
E(X) = 1232.
Variance:
To find the variance of X, we first need to calculate the squared deviation of each outcome from the expected value. For each outcome, subtract the expected value from the outcome and square the result.
In this case, the possible outcomes are 0, 1, 2, 3, and 4, with corresponding probabilities P(X = 0), P(X = 1), P(X = 2), P(X = 3), and P(X = 4) calculated earlier.
The squared deviation for each outcome can be calculated as (outcome - expected value)^2. For example, for the outcome X = 0, the squared deviation is (0 - 1232)^2 = 1521024.
Repeat this calculation for each outcome and multiply each squared deviation by its corresponding probability. Finally, sum up all the squared deviations multiplied by their probabilities to get the variance. The formula for the variance is
Var(X) = (0 - E(X))^2 * P(X = 0) + (1 - E(X))^2 * P(X = 1) + (2 - E(X))^2 * P(X = 2) + (3 - E(X))^2 * P(X = 3) + (4 - E(X))^2 * P(X = 4).
Calculating this gives us
Var(X) = (0 - 1232)^2 * 1820 + (1 - 1232)^2 * 728 + (2 - 1232)^2 * 28 + (3 - 1232)^2 * 56 + (4 - 1232)^2 * 70
Var(X) = 17,681,776 + 4,495,296 + 65,024 + 12,544 + 25,200
Var(X)= 22,279,840.