Practice question's part b, (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 0 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 3. 6. Probability e (1 – 20) (1- 20) (1- 20) (1- 20) 0 (ii) The value of 0 is unknown. Data from which to estimate the value of 0 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 4. 6. Frequency 205 154 141 165 145 190 Show that the likelihood of 0 based on these data is L(0) = Ce395 (1 – 20)605, where C is a positive constant, not dependent on 6. (iii) Show that L'(0) = C6394 (1 – 20)04 (395 – 2000 0). (iv) What is the value of the maximum likelihood estimate, 6, of 0 based on these data? Justify your answer. What does the value of 0 suggest about the value of 0 for this biased die compared with the value of 0 associated with a fair, unbiased, die?
Practice question's part b, (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 0 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 3. 6. Probability e (1 – 20) (1- 20) (1- 20) (1- 20) 0 (ii) The value of 0 is unknown. Data from which to estimate the value of 0 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 4. 6. Frequency 205 154 141 165 145 190 Show that the likelihood of 0 based on these data is L(0) = Ce395 (1 – 20)605, where C is a positive constant, not dependent on 6. (iii) Show that L'(0) = C6394 (1 – 20)04 (395 – 2000 0). (iv) What is the value of the maximum likelihood estimate, 6, of 0 based on these data? Justify your answer. What does the value of 0 suggest about the value of 0 for this biased die compared with the value of 0 associated with a fair, unbiased, die?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Practice question's part b,
(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 0 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
3
6.
Probability 0 (1 – 20) (1– 20) (1- 20) (1– 20) 0
(ii) The value of 0 is unknown. Data from which to estimate the value
of 0 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
Frequency 205
154
141
165
145
190
Show that the likelihood of 0 based on these data is
L(0) = C 03905 (1 – 20)605,
where C is a positive constant, not dependent on 6.
(iii) Show that
L'(0) = C 6394 (1 – 20)604 (395 – 2000 0).
(iv) What is the value of the maximum likelihood estimate, 6, of 0
based on these data? Justify your answer. What does the value of
0 suggest about the value of 0 for this biased die compared with the
value of 0 associated with a fair, unbiased, die?
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