The markov chains

Stats 125 - Tutorial 10 Semester 2, 2019 1. Discussion and Quiz question Two dogs, Hinkler and Moke, have three fleas between them. Every day, one (and only one) flea jumps from the dog it is currently on and lands on the other dog. On each day, which flea jumps is random, and all three fleas have the same probability of jumping, regardless of which dogs the fleas are on. Define X, to be the number of fleas on Moke after n days. Then (X,, n = 0, 1, 2, . ..) forms a Markov chain with state space S = {0, 1, 2, 3}. Initially, all three fleas are on Hinkler, meaning that Xo = 0. (a) Some of the entries of the transition matrix, P, are shown. Find the rest. 0 P = 2/3 0 1/3 (b) Confirm to yourself that this Markov chain is a birth-death chain. (c) Find the equilibrium distribution(s) # = (#o, #1, #2, #3). Is there a unique equilibrium distribution, or more than one? Hint: use the Detailed Balance Equations.1 Normal 1 No Spac... Heading 1 Heading 2 Title Subtitle Subtle Er... Emphasis Intense E Styles BP Correlation and Regression 10 bowlers were randomly selected and were asked how long they have bowled (in years] and their current bowling average. The results are below: Bowler # 2 3 5 6 7 8 9 10 Years bowling (x): 10 5 20 6 5 4 Current average (v): | 70 185 165 120 195 200 150 170 150 135 Computer by hand and check your results with the calculator. Perform a t-test for the correlation coefficient using a = 0.05 Find the linear regression model for this data. (Check g and b on your calculator.) Use your model to predict the Bowling average of a person with 12 years of experience. Use your model to predict the years of experience for a bowler with a 175 bowling average.1. Consider the simple linear regression model Y: = a+l31$e +61: Where the errors a are identically and independently distributed as N(0,02). (a) If the predictors satisfy :7: = 0, show that the least squares estimates 50 and 31 are independently distributed. (in) Let r be the sample correlatiOn coeicient between the predictor and response. Under what conditions will we have 51 = r? (0) Suppose that Bl = r, as_in part b}, but make no assumptions on i. Give the tted regression equatic-n (in terms of Er, Y and r) and use it to show that Iii 5 I23 |. This inequality helps_explain the meaning of the term 'Eegression" since the tted value 1% is closer to the average Y than the predictor 93.; is to 5:, Le. it regresses toward the mean