Question: From New York State Department of Motor Vehicles (DMV) records, it is known that 85% of all vehicle registered in New York State have automatic transmissions, and the rest manual transmissions. It is also known that 70% of vehicles with automatic transmission are classified as passenger cars and the rest as trucks, while 45% of manual transmission cars are passenger cars and the rest trucks. Using this information, answer the following questions.
a) Find the joint distribution of vehicle type (passenger car / truck) and transmission type (automatic / manual) in New York.
b) Find the marginal distribution of vehicle type in New York.
c) Find the conditional distribution of vehicle type given that the transmission is automatic.
d) Find the conditional distribution of vehicle type given that the transmission is manual.