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 understand the probability distribution of X.
Let's analyze the problem step by step:
1. Determine the probability distribution:
- The boat can be located at any point uniformly distributed inside the circular pool of radius 1 mile.
- Since the bird lands on the rope, its location can be anywhere along the rope.
- The rope is a segment connecting the center of the pool to the boat, forming a circle of radius 1 mile.
2. Find the random variable X:
- X represents the distance between the center of the pool and the bird.
- As the bird lands on the rope, X corresponds to the length of the rope.
3. Compute the expected value (E[X]):
- The expected value of X, denoted as E[X], represents the average distance between the center of the pool and the bird.
- Since the bird can land anywhere along the rope, E[X] is calculated as the average length of the rope.
- In this case, the length of the rope is the radius of the circle, which is 1 mile.
- Therefore, E[X] = 1 mile.
4. Compute the variance (var(X)):
- The variance of X, denoted as var(X), measures the spread or variability of X.
- To calculate var(X), we need to calculate the expected value of X squared (E[X^2]) and subtract the square of the expected value (E[X]).
- Calculate E[X^2]:
- The bird can land anywhere along the rope, so X^2 represents the squared length of the rope.
- The square of the length of the rope is simply the square of the radius, which is 1 mile.
- Hence, E[X^2] = 1^2 mile^2 = 1 mile^2.
- Calculate var(X):
- var(X) = E[X^2] - (E[X])^2
- Plugging in the values: var(X) = 1 mile^2 - (1 mile)^2 = 1 - 1 = 0 mile^2.
So, the expected value E[X] is 1 mile, and the variance var(X) is 0 mile^2.