Question :

Suppose that X and Y have a discrete joint distribution for which the joint probability function is: k(ty) for x = 0, 1, 2 and y = 0, 1, 2, 3 p(x, y) otherwise. i. Determine the value of the constant, k. ii. Determine the marginal distributions of X and Y, respectively. iii. Are X and Y independent? Explain your answer. (20 marks)we have k = 1/30. ii. For the marginal distribution of X we have: Px(x) = (2x + 3)/15 for x = 0, 1, 2 otherwise. For the marginal distribution of Y we have: PY (y) = (y + 1)/10 for y = 0, 1, 2, 3 0 otherwise.(b) i. A coffee machine can be calibrated to produce an average of u millilitres (ml) per cup. Suppose the quantity produced is normally distributed with a standard deviation of 9 ml per cup. Determine the value of u such that 250 ml cups will overflow only 1% of the time. (4 marks) ii. Suppose now that the standard deviation, o, can be fixed at specified levels. What is the largest value of o that will allow the amount of coffee dispensed to fall within 30 ml of the mean with a probability of at least 95%? (4 marks)