A Safety Officer for an auto insurance company in CA was interested in learning how the extent of an individual's injury in an automobile accident relates to the type of safety restraint the individual was wearing at the time of the accident. As a result, the Safety Officer used statewide ambulance and police records to compile the following table: None Belt only Belt and Harness None 0.06 0.075 0.065 Minor 0.135 Major 0.135 Death 0.025 0.21 0.115 0.1 0.065 0.01 0.015 For the sake of understanding the Safety Officer's terminology, let's assume that "Belt only" means that the person was only using the lap belt, whereas "Belt and Harness" should be taken to mean that the person was using a lap belt and shoulder strap.
Also, note that the Safety Officer created the random variable X, the extent of injury, by arbitrarily assigning values 0, 1, 2, and 3 to each of the possible outcomes None, Minor, Major, and Death. Similarly, the Safety Officer created the random variable Y, the type of restraint, by arbitrarily assigning values 0, 1, and 2 to each of the possible outcomes None, Belt Only, and Belt and Harness.
(a) What is the probability a randomly selected person in an accident was wearing a seat belt and had only a minor injury?
(b) If a randomly selected person wears no restraint, what is the probability of death?
(c) If a randomly selected person sustains no injury, what is the probability the person was wearing a seat belt and harness? (d) Rewrite f(r, y) in a mathematical expression.
(e) Derive the marginal distribution fx(x) and fy (y), and draw each marginal distribution.
(f) Compute ux, HY, ox, and oy (Hint: SD(X) = Var(X)).
(g) Compute uxY-2
(h) Compute Ayx-2-
(i) Compute Cov(X, Y)and discuss the relationship between X and Y.
(j) Compute Cor(X, Y)and discuss the magnitude of the relationship between X and Y. %3D