1.A random variable Y , taking values in R+, is said to have a lognormal distribution with parameters μ and σ2 if lnY ∼ N(μ,σ2). Find the probability density function of Y .6 marks
2.Suppose that X1 and X2 are independent discrete random variables. Show that the covariance between them is zero.4 marks
3.A television quiz game operates as follows. In the first part of the game, a contestant isasked a series of difficult questions; the probability of answering any question correctly is p, independently of all other questions. This first part of the game ends as soon as the contestant answers a question incorrectly. Denote by N the total number of questions answered correctly in this first part of the game (excluding the final incorrect one). In the second part of the game, the contestant is asked a series of easy questions; here, the probability of answering any question correctly is r, independently of all other questions. The total number of questions asked in this second part of the game is N (i.e. if N = n in the first part of the game, the contestant answers n questions in the second part). The contestant’s final score, Y say, is equal to the number of correct answers in the second part of the game.(a)Find the probability mass function of N. Also state the conditional distribution of Y given N = n, including the value(s) of its parameter(s).(b)Use the Law of Total Probability to find the probability mass function of Y . Show that this pmf has the same form as thatof N, but with p replaced by another quantity which you should define.Hint: you are reminded of the series expansion (1−x)−m = 1+mx+m(m+1)x2/2+…, for |x| < 1.10 marks
