[1 mark] ii. Find the median of X . [2 marks] iii. Deduce the value of c given that E ( X + c ) = 3 E ( X - c ). [1 mark] iv. Two independent observations of X are taken. Find the probability that one is less than 1 2 and the other is greater than 2. [2 marks] v. Find the cumulative distribution function (cdf) of X . [2 marks] 1
(b) The time, T years, before a machine in a factory breaks down follows the probability density function given by f ( t ) = braceleftBigg ate - bt t > 0 0 otherwise where a and b are positive constants. It may be assumed that, if n is a positive number, integraldisplay ∞ 0 t n e - bt dt = n ! b n +1 . The mean of T is known to be 1.5. Find the values of a and b . [2 marks] Note: n ! is read ”n factorial”. For example, 5! is 120 because 5! = 5 X 4 X 3 X 2 X 1. n ! is just shorthand for n X ( n - 1) X ( n - 2) X .... 3 X 2 X 1. You should be able to find this function in your scientific calculator. 3. The distance by road from UBC Vancouver to UBC Okanagan is 400 km. When I travel by car, my average speed is V km/hr, where V is uniformly distributed over the interval 60 ≤ V ≤ 80. (a) Write down the cumulative distribution function of V . [2 marks] (b) Obtain the cumulative distribution function of T , the time in hours for the journey from UBC Vancouver to UBC Okanagan. Hence, obtain the probability density function of T . [5 marks] (c) Calculate E ( T ) and var ( T ). [2 marks] (d) I have to attend a meeting at UBC Okanagan which starts at 2 pm. Find the latest time I can leave UBC Vancouver in order that the probability of my arriving in time for the meeting should be at least 80%.