STATISTICS 111 HOMEWORK 5 - due at recitation classes on 10/14/2016. NOTE 1. Because of the mid-term break there will not be a recitation class on October 7. Thus this homework will be due at recitation class on October 14. Also, because you have two weeks to do this homework, it is slightly longer than usual and carries more points than usual. NOTE 2. The theme of these questions will be continued later, so keep a copy of your answers. NOTE 3. Some of the questions below refer to Question 1 on Homework 4. Thus this question is repeated here for your convenience. Suppose that a fair die is to be rolled twice. In Homework 3, Question 1, you found the probabilities that the sum of the two numbers to appear (which we denote T2 ) will be (a) 4, (b) 5, and (c) 6. Now extend the reasoning that you used to the possible values T2 = 4, 5 and 6 to find the probabilities that T2 takes the respective values (a) 2, (b) 3, ...., (k) 12. Also, use the \"symmetry\" property of a mean of a symmetric distribution to find the mean of T2 immediately. ********************************** Many statistical procedures require us to consider sums, averages and differences of random variables. Questions 1, 2, 3 and 4 consider a sum, an average and a difference. Here is some relevant theoretical background as given in class and also in the handout notes. A fair die is to be rolled twice. We denote the number that will turn up on roll 1 by X1 and the number that will turn up on roll 2 by X2 . Then the sum of these random variables is defined as T2 , so that T2 = X1 + X2 . As discussed in class, this sum is a random variable, since it is the sum of two random variables. We define the average of these random variables by 12 (X1 + X2 ). We denote this average by - pronounced \"X-bar\". This average is also a random variable. This is in contrast to a X mean, which is a parameter. For the purposes of the questions below, we define D by D = X1 X2 . (So D is the number that will turn up on the first roll minus the number that will turn up on the second roll. 1 This means that some of the possible values of D are negative.) This difference is also a random variable, since it is defined in terms of random variables. NOTES for questions 1, 2, 3 and below. (a). We assume in Questions 1, 2, 3 and 4 that the random variables X1 and X2 are iid, that is they are identically and independently distributed. (b). We know from class that the mean of X1 is 3.5 and the variance of X1 is 35/12. The same values apply for X2 . (c). Present all your probabilities in fractional form (i.e. in the form a/b, where a and b are integers (i.e. whole numbers)), not in decimal form such as 0.wxyz. Also, use the same value of the denominator b for all the probabilities and other quantities that you calculate. Also, present all the means and variances asked for in this homework in fractional form a/b, where a and b are integers. In other words, always do exact calculations. Never use decimals. ********************************** 1(i). Use the numerical values in Note (b) above, together with Equations (24) in the handout notes, to find the mean and the variance of T2 . [4 points] 1(ii). From your answer to Question 1 in Homework 4, use the relevant \"long\" formulas to calculate the mean and the variance of T2 . (Show the details of your calculations.) Do your values agree with the values that you found in part (i) of this question? [4+5+1 = 10 points] 2. From your answer to Question 1 of Homework 4, find (quickly) and write down in That is, write down a table showing all \"tableau\" form the probability distribution of X. together with their associated probabilities. (Hint: Note that the possible values of X, 1 = 4.5. Thus 4.5 is a possible value of X, X = 2 T2 . Thus if for example T2 = 9, then X = 4.5 is the same as the probabilitiy that T2 = 9.) and the probability that X [8 points] 3(i). Use the numerical values in Note (b) above, together with Equations (25) in the hand out notes, to find the mean and the variance of X. [4 points] 3(ii). From your answer to Question 2, use the relevant \"long\" formula calculations to find (Show all the details of your calculations.) Do your values the mean and variance of X. agree with the values that you found in part (i) of this question? [6 points] 4(i). We define D as X1 X2 . Find and write down in \"tableau\" form the probability distribution D. That is, write down a table showing all the possible values of D, together with 2 their associated probabilities. (This will involve some tedious work, using however nothing more than \"common sense\". For example D will equal -2 if X1 = 1 and X2 = 3, or if X1 = 2 and X2 = 4, or if X1 = 3 and X2 = 5, or if X1 = 4 and X2 = 6.) From your answer use the relevant \"long\" formulas to find the mean and variance of D. (Show all the details of your calculations.) [7 points] 4(ii). Use the numerical values in Note (b) above, together with Equations (26) in the handout notes, to find the mean and the variance of D. Do your values agree with the values that you found in part (i) of this question? [2+2+1 = 5 points] 5. Consider the random variable T4 , the sum of the numbers turning up on FOUR rolls of a fair die. From your answer to Homework 4, Question 1, find the probability distribution of T4 . (This will involve some tedious work, using however nothing more than \"common sense\". Think of T4 as Y1 + Y2 , where Y1 is the sum of the numbers turning up on rolls 1 and 2 of the die, and Y2 is the sum of the numbers turning up on rolls 3 and 4 of the die. Now think (for example) of the ways in which T4 can equal 15. One way that this could happen would be if Y1 = 3 and Y2 = 12. Another way that this could happen would be if Y1 = 4 and Y2 = 11. And so on.) [8 points] 6. Use the long formulas for the mean and a variance to find the mean and variance of T4 (as defined in Question 5), using the results of your calculations in Question 5. [3+4=7 points] 7. Use the short formulas for the mean and the variance of a sum (see Equations (24) in the handout notes) to find the mean and variance of T4 quickly. Do these agree with the values that you calculated in Question 6? [5 + 1 = 6 points] TOTAL: 65 points 3