8. [15 marks] A fair sixsided die is rolled and a fair coin is tossed. A head counts for 1 and a tail for 0. The die face and the 'value' of the coin toss are added to form the random variable X. (eg. rolling a 3 and tossing a head gives X = 4.) (a) Write down the sample space S X for X. (b) Calculate the PMF for X. (A table might help here.) (c) Calculate the value of the CDF for X at X = 4.5 (d) Find the expected value of X, E'(X). (0) Calculate the probability that X is even given that it is less than seven. 10. [9 marks] When a conventional paging system transmits a message1 the probability that the message will be received by the appropriate pager is p. To be condent that a message is received at least once, a system transmits the message 11 times. (a) Assuming all transmissions are independent, what is the PMF of K, the number of times the pager receives the same message? (b) Assume p=0.7. What is the minimum value of n that produces a probability of 0.95% of receiving the message at least once? (c) Assume p = 0.7. Assume also that a successful transmission costs $2.00 and an unsuccessful one costs $0.25. The cost of a page Y is then a random variable. Describe the sample space and nd PMF of Y