A boat is located at a random location uniformly distributed inside of the circular

pool of radius 1 mile. The boat is tied by a rope to a pole located directly at the center of the
pool. A bird lands on the rope at the uniformly random location. If X is the distance between
the center of the pool and the bird compute E[X] and var(X).

To compute the expected value (E[X]) and variance (var(X)) of the distance (X) between the center of the pool and the bird, we need to utilize basic concepts from probability and statistics.

First, let's consider the probability distribution of the random variable X. Since the bird lands on the rope at a uniformly random location, this implies that the probability of the bird landing at any particular distance from the center of the pool is uniformly distributed.

Given that the boat is located uniformly inside the circular pool of radius 1 mile, we can use geometric considerations to determine the probability distribution of X.

Consider that X represents the distance between the center of the pool and the bird. In terms of X, we have the following possible scenarios:
1. If X ≤ 1, the bird is on the rope and within the circular pool.
2. If X > 1, the bird is outside the circular pool.

Since X is uniformly distributed, the probability of the bird landing within the pool (X ≤ 1) is given by the ratio of the area of the pool to the total area.

The area of a circular pool with a radius of 1 mile is π * (1 mile)^2 = π square miles.

Hence, the probability of the bird landing within the circular pool is P(X ≤ 1) = Area of Pool / Total Area = (π square miles) / (π * (1 mile)^2) = 1.

Now, let's compute the expected value (E[X]) and variance (var(X)) of X.

1. Expected Value (E[X]):
The expected value of X represents the average distance between the center of the pool and the bird. Since X is uniformly distributed, the expected value is given by the midpoint of the interval [0, 1]. Therefore, E[X] = (0 + 1) / 2 = 1/2 mile.

2. Variance (var(X)):
The variance of X is a measure of the spread or dispersion of X around the expected value. For a uniform distribution, the variance is given by (1/12) * (b - a)^2, where a and b are the endpoints of the distribution range.

In this case, the range of X is [0, 1], so the variance of X is (1/12) * (1 - 0)^2 = 1/12 square miles.

To summarize:
E[X] = 1/2 mile (expected value of X)
var(X) = 1/12 square miles (variance of X)