(b) An otherwise fair siy eided die has been tamnpered 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 2 3 4 6. Probability (1- 20) (1– 20) (1- 20) (1 – 20) 0 Heeessaryfo (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 6. Frequency 205 154 141 165 145 190 Show that the likelihood of 0 based on these data is L(0) = C 6395 (1 – 20)605, where C is a positive constant, not dependent on 0. (iii) Show that State what rule is ased, L'(0) = C 64 (1 – 20)604 (395 – 2000 6). (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 O suggest about the value of 0 for this biased die compared with the value of 0 associated with a fair, unbiased, die?
(b) An otherwise fair siy eided die has been tamnpered 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 2 3 4 6. Probability (1- 20) (1– 20) (1- 20) (1 – 20) 0 Heeessaryfo (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 6. Frequency 205 154 141 165 145 190 Show that the likelihood of 0 based on these data is L(0) = C 6395 (1 – 20)605, where C is a positive constant, not dependent on 0. (iii) Show that State what rule is ased, L'(0) = C 64 (1 – 20)604 (395 – 2000 6). (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 O 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
Problem 1P
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Transcribed Image Text:(6) An otherwise fair sir 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 ở 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
0 (1- 20) (1 – 20) (1– 20) {(1 – 20) 0
Heeessary
(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
3
4 5
6.
Frequency 205
154
141
165
145
190
Show that the likelihood of 0 based on these data is
L(0) = C0395 (1 – 20)605,
where C is a positive constant, not dependent on 0.
(iii) Show that
Stare what rule is ased,
L'(0) = C 6394 (1 – 20)®04 (395 – 2000 6).
(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
O 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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