Question

In a football training, the player is kicking the ball towards a round-shaped object. First he weeks his coach to decide the radius of the round object, X, which has a exponential distribution with parameter $2\lambda$. After this, He kicks the football towards a random point inside the round object (assuming that the ball is so small with respect to the object that the ball can hit every point on the object). Define random variable R as the distance between the hitting point and the center of the round object.

1) Find the expectation E(R)

2) Find Cov(X,R)

Answer 1