Consider the hat problem, with n=10. What is the expected value of X3X6X7?
To find the expected value of X3X6X7 in the hat problem, we need to determine the probability distribution of X3, X6, and X7 and then calculate their respective expected values.
In the hat problem, there are n=10 distinct hats labeled from 1 to 10. Let's assume that these hats are randomly distributed among n people, numbered from 1 to n. Each person selects a hat blindly and independently of others.
First, let's determine the probability distribution for X3, X6, and X7 individually.
X3 represents the outcome of the third person's hat selection. Since each person selects a hat uniformly at random, the probability that the third person chooses a specific hat is 1/n. Therefore, the probability distribution for X3 can be represented as P(X3=k) = 1/n, where k is a hat number from 1 to n.
Similarly, the probability distribution for X6 and X7 can be represented as P(X6=k) = 1/n and P(X7=k) = 1/n, respectively.
Now, to find the expected value of X3X6X7, we need to calculate the product of the expected values of X3, X6, and X7.
The expected value of a discrete random variable is calculated by summing the product of each possible outcome with its corresponding probability.
For X3, the expected value E(X3) is calculated as:
E(X3) = Σ (k * P(X3=k))
Since each hat is equally likely to be chosen, the sum can be simplified as:
E(X3) = (1/n) * (1 + 2 + 3 + ... + n)
Using the formula for the sum of an arithmetic series, (n * (n + 1)) / 2, we can simplify the expression further:
E(X3) = (1/n) * ((n * (n + 1)) / 2)
Similarly, the expected value for X6 and X7 would also be (1/n) * ((n * (n + 1)) / 2).
Now, to find the expected value of X3X6X7, we multiply these three expected values together:
E(X3X6X7) = E(X3) * E(X6) * E(X7)
Substituting the expressions:
E(X3X6X7) = (1/n) * ((n * (n + 1)) / 2) * (1/n) * ((n * (n + 1)) / 2) * (1/n) * ((n * (n + 1)) / 2)
Simplifying the expression further:
E(X3X6X7) = ((n * (n + 1)) / 2)^3 / n^3
So, the expected value of X3X6X7 in the hat problem, with n=10, is ((10 * 11) / 2)^3 / 10^3, which can be further simplified to 11^3 / 8, or 1331/8.
Therefore, the expected value of X3X6X7 is 166.375.