A4.1 Let X N U(0,1) be a uniformly distributed random variable with parameters 0 and 1. Let Z = log(1 X), so that Z is a random variable taking values in (0,00). (i) For arbitrary z > 0. find P(Z g 2). (ii) Using your answer in (i) find the probability density function of Z. What is the distribution of Z? A42 Let T1 and T2 be independent continuous random variables with T1 ~ Exponentia|(}\\) and T2 ~ Exponentialu). Let T = min(T1,T2). Express the event {T > t} as an intersection of two events involving T1 and T2. Hence find PC!" > t). [Hintz independence] What is the distribution of T? Every question is worth 3 marks, awarded as follows: 3 - essentially perfect: 2 - minor mis- takes/gaps/inaccuracies; 1 some good ideas but with major mistakes and incomplete: O - entirely wrong or no answer at all or wrong question. An additional mark is awarded if the writing style is clear, readable and accurate and the paper looks overall clean. This results in a maximum of 7 marks for this coursework