MATH 5305: Assignment 2 Due Date: October 12, 2015 (at the start of the class) 1. A manufacturer of electronic component is interested in determining the lifetime of a certain type of battery. A sample, in hours of life, is as follows: 123, 116, 122, 110, 175, 126, 125, 111, 118, 117 (a) Find the sample mean and sample variance [6 marks] (b) Obtain the ve number summary [8 marks] (c) Draw a box plot and comment on the shape of the distribution of the lifetimes of battery. Give reasons when commenting on the shape [9 marks] (d) Identify any potential outliers by proper reasoning [5 marks] 2. A coin is biased such that a head is three times as likely to occur as a tail. Find the expected number of tails when the coin is tossed twice [5 marks] 3. Let X be a random variable with the following probability distribution: -3 X P[X=x] 1/6 6 1/2 9 1/3 (a) Find E(X) and E(X 2 ) [6 marks] (b) Find V (X) [3 marks] (c) Let Y = (2X + 1)2 . Find E(Y ) using (a) [6 marks] (d) From E(X) and V (X), what can you say about the distribution of X [2 marks] 4. By investing in a particular stock, a person can make a prot of $4000 in a year with probability 0.3 or take a loss of $1000 with probability 0.7. What is this persons expected gain [5 marks] 5. If a random variable X is dened such that E[(X 1)2 ] = 10 and E[(X 2)2 ] = 6. Find E(X) and V (X) [8 marks] 6. The probability that a patient recovers from a heart operation is 0.9. If 10 patients go through this heart operation, how many patients do you expect to recover [5 marks] 7. In a survey, 656 students, who were 13 to 18 years old, were asked about adequate sleep and exercise behaviors. The data is represented below in a 2 2 table which gives the counts with students classied as getting or not getting adequate sleep and by exercise level Enough Sleep Yes No 1 Exercise Level High Low 151 115 148 242 (a) Find the distribution of enough sleep for high exercisers [3 marks] (b) Repeat (a) for low exercisers [3 marks] (c) Find the distribution of exercise level for those who get enough sleep [3 marks] (d) Repeat (c) for those who do not get enough sleep [3 marks] (e) Using the results of (a) and (b), summarize the relationship between enough sleep and exercise level [5 marks] (f) Using the results of (c) and (d), summarize the relationship between enough sleep and exercise level [5 marks] (g) Compare the results in (e) and (f) and comment [4 marks] 8. Let X and Y be two correlated random variables. If the correlation coecient between X and Y is 1, i.e., (X, Y ) = 1, show that V (X + Y ) = V (X) + V (Y ) [6 marks] 2