5. (a) Assume that the random variable Y follows a normal distribution N(, a + 1). Assume a is the last digital number of your student id number (for example, if your student number is 20987654D, then a = 4). [5 marks] (i) Write down your student number and variance 2 of this normal distribution; (ii) If the probability that the quadratic equation a2+8x + Y = 0 has no real root is 0.6664, find the value of p; (iii) Furthermore, find the probability that the equation has exactly one solution. 2 (b) Assume that the score of one standard test follows a normal distribution with μ = 73 and σ = 4. If all candidates are formed into many different groups of 36 individuals, will a group with average mark 85 be in the top ten percent of all groups with highest group average marks? [5 marks] (c) It is known that a machine produces defects at a probability of 5%. Find the probability that more than 40 defects are produced out of 500 items produced by this machine. [5 marks] (d) An unfair die has four faces numbered 0, 1, 2, 3, and the probability of getting the corresponding score is shown in the following table face (score) 01 2 3 probability 0.1 0.2 0.3 0.4 Assume this die is rolled 90 times. Determine the probability that the sum of the 90 scores obtained will exceed 200. [5 marks]