Answer:
the expected value of the distance between X and Y in this case is 0.
Explanation:
(a) When both X and Y are selected independently from the range 0-1, the distance between the two points can be calculated as the absolute difference between X and Y, denoted as |Y - X|. To find the expected value of this distance, we need to integrate over all possible values of X and Y and weight them by their respective probabilities.
The probability density function (PDF) for a uniform distribution on the interval 0-1 is given by f(x) = 1 for 0 ≤ x ≤ 1, and 0 otherwise.
Therefore, the expected value of the distance between X and Y in this case can be calculated as:
E(|Y - X|) = ∫∫ |y - x| f(x) f(y) dx dy
= ∫∫ |y - x| dx dy
= ∫ [∫ |y - x| dx] dy
Evaluating the inner integral:
∫ |y - x| dx = (x - y) / 2 | 0 ≤ x ≤ y + (y - x) / 2 | y ≤ x ≤ 1
= (y - y) / 2 + (1 - y) / 2
= (1 - y) / 2
Substituting this result back into the outer integral:
E(|Y - X|) = ∫ (1 - y) / 2 dy | 0 ≤ y ≤ 1
= [(1 - y)^2 / 4] | 0 ≤ y ≤ 1
= (1 - 1/4) - (1 - 1) / 4
= 3/8
Therefore, the expected value of the distance between X and Y when both are selected independently from the range 0-1 is 3/8.
(b) In this case, we select X between 0 and 1 and then select Y between 0 and X. The distance between the two points is given by Y - X.
To find the expected value, we need to integrate over the joint probability density function (PDF) of X and Y.
The joint PDF of X and Y can be expressed as:
f(x, y) = f(x) f(y|x) = 1 * 1/x = 1/x, for 0 ≤ x ≤ 1 and 0 ≤ y ≤ x
Now we can calculate the expected value of the distance:
E(Y - X) = ∫∫ (y - x) f(x, y) dx dy
= ∫∫ (y - x) (1/x) dx dy
Integrating the inner integral:
∫ (y - x) (1/x) dx = (y - x) ln(x) | 0 ≤ x ≤ y
= (y - y) ln(y) - (y - 0) ln(0) | 0 ≤ y ≤ 1
= 0
Therefore, the expected value of the distance between X and Y in this case is 0.
To summarize:
a) When X and Y are selected independently from the range 0-1, the expected value of the distance |Y - X| is 3/8.
b) When X is selected between 0 and 1, and Y is selected between 0 and X, the expected value of the distance Y - X is 0.