A bag contains 4 balls, each of which have a positive integer written on it. Let X be the random variable of the value written on a single ball drawn from the bucket. If E[X]=8, what is the maximum possible value of E[X^2]?
To find the maximum possible value of E[X^2] given that E[X] = 8, we need to consider the relationship between the mean (expected value) and the second moment of a random variable.
The mean (E[X]) is calculated by taking the sum of all possible values of X multiplied by their respective probabilities. E[X^2] represents the second moment of X and is calculated by taking the sum of all possible values of X^2 multiplied by their respective probabilities.
Let's denote the four positive integer values written on the balls as a, b, c, and d. Since we want to find the maximum possible value of E[X^2], we need to maximize the individual values of a^2, b^2, c^2, and d^2.
Since E[X] = 8, we know that:
(Equation 1) (a + b + c + d) / 4 = 8
We want to maximize the values of a^2, b^2, c^2, and d^2, given that Equation 1 holds.
To do this, we can use the AM-GM (Arithmetic Mean - Geometric Mean) inequality, which states that for any set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean.
Applying the AM-GM inequality to a + b + c + d, we have:
(a + b + c + d) / 4 ≥ √(abcd)
Since the AM-GM inequality holds, we can substitute √(abcd) with (abcd)^(1/4). Therefore, we can rewrite the inequality as:
(a + b + c + d) / 4 ≥ (abcd)^(1/4)
To satisfy Equation 1, we can set (a + b + c + d) / 4 = 8. Substituting this into the inequality, we get:
8 ≥ (abcd)^(1/4)
To find the maximum value of (abcd)^(1/4), we need to maximize the value of abcd.
Since a, b, c, and d are positive integers, we can maximize their values by setting them as close to each other as possible. So we set a = b = c = d.
Substituting these equal values into the equation, we get:
8 ≥ (abcd)^(1/4)
8 ≥ (aaaa)^(1/4)
8 ≥ a^(4/4)
8 ≥ a
This equation implies that the maximum value for a (and consequently b, c, and d) is 8.
Therefore, the maximum possible value of E[X^2] is obtained when a = b = c = d = 8. In this case, E[X^2] = E[8^2] = E[64] = 64.