VEX 2 BW A+ 0 X X 64 CA https://learn2.open.ac.uk/63/b2/63b253ee... + 3 of 5 (b) An otherwise fair six-sided die has been tampered with in an attempt to cheat at a dice game. The effect is that the 1 and 6 faces have a different probability of occurring than the 2, 3, 4 and 5 faces. Let be the probability of obtaining a 1 on this biased die. Then the outcomes of rolling the biased die have the following probability mass function. Table 1 The p.m.f. of outcomes of rolls of a biased die A Outcome 1 2 3 4 5 6 Probability (1-20) (1-20) (1-20) (1-20) 0 (i) By consideration of the p.m.f. in Table 1, explain why it is necessary for to be such that 0 < 0 < 1/2. (ii) The value of is unknown. Data from which to estimate the value of ◊ were obtained by rolling the biased die 1000 times. The result of this experiment is shown in Table 2. Table 2 Outcomes of 1000 independent rolls of a biased die Outcome 1 2 3 45 6 Frequency 205 154 141 165 145 190 Show that the likelihood of 0 based on these data is [2] L(0) = C 0395 (1 - 20)605, 111 where C is a positive constant, not dependent on 0. Show that [5] I'(0) C0394 (1-20)604 (395 - 20000). [4] (iv) What is the value of the maximum likelihood estimate, 0, of 0 based on these data? Justify your answer. What does the value of 0 suggest about the value of 6 for this biased die compared with the value of 0 associated with a fair, unbiased, die? [4] B 05:01 17/12/2024 F9 F10 F11 F12 IA Prt Sc Insert Delete F7 & 7 F8 U 8 O P + Backspace #

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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(b) An otherwise fair six-sided die has been tampered with in an attempt to cheat at a dice game. The effect is that the 1 and 6 faces have a different probability of occurring than the 2, 3, 4 and 5 faces. Let θ be the probability of obtaining a 1 on this biased die. Then the outcomes of rolling the biased die have the following probability mass function. Table 1 The p.m.f. of outcomes of rolls of a biased die Outcome 1 2 3 4 5 6 Probability θ 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) θ (i) By consideration of the p.m.f. in Table 1, explain why it is necessary for θ to be such that 0 < θ < 1/2. [2] (ii) The value of θ is unknown. Data from which to estimate the value of θ were obtained by rolling the biased die 1000 times. The result of this experiment is shown in Table 2. Table 2 Outcomes of 1000 independent rolls of a biased die Outcome 1 2 3 4 5 6 Frequency 205 154 141 165 145 190 Show that the likelihood of θ based on these data is L(θ) = C θ395 (1 − 2θ) 605 , where C is a positive constant, not dependent on θ. [5] (iii) Show that L ′ (θ) = C θ394(1 − 2θ) 604 (395 − 2000 θ). [4] (iv) What is the value of the maximum likelihood estimate, θb, of θ based on these data? Justify your answer. What does the value of θb suggest about the value of θ for this biased die compared with the value of θ associated with a fair, unbiased, die? [4]