![2 Maximum Likelihood [25pts] 2.1 Discrete Example 17pts) Suppose you are playing two unfair coins. The probability of tossing a tail is θ for coin 1, and 20 for coin 2. You toss each coin for several times, and you get the following results: (a) What is the probability of tossing a head for coin 1 and for coin 2 3pts? Coin No. eult head head tail head tail tail (b) what is the likelihood of the data given θ [7pts? (c) What is maximum likelihood estimation for θ [7pts](http://img.homeworklib.com/questions/424a0260-77e5-11ea-b840-2dd6597f4516.png?x-oss-process=image/resize,w_560)
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2 Maximum Likelihood [25pts] 2.1 Discrete Example 17pts) Suppose you are playing two unfair coins. The...
Suppose you are playing two unfair coins. The probability of tossing a tail is θ for...
Suppose you are playing two unfair coins. The probability of
tossing a tail is θ for coin 1, and 2θ for coin 2. You toss each
coin several times, and you get the following results:
(a) What is the probability of tossing a head for coin 1 and for
coin 2 [3pts]?
(b) What is the likelihood of the data given θ [7pts]?
(c) What is maximum likelihood estimation for θ [7pts]?
Coin No.Result head head tail head tail tail
Suppose you are playing two unfair coins. The probability of tossing a tail is θ for...
Suppose you are playing two unfair coins. The probability of tossing a tail is θ for coin 1, and 2θ for coin 2. You toss each coin for several times, and you get the following results: (a) What is the probability of tossing a head for coin 1 and for coin 2? coin no. result 1 head 2 head 1 tail 1 head 2 tail 2 tail (b) What is the likelihood of the data given ? (c) What is...
Suppose you toss an unfair coin 8 times independently. The probability ofgetting a head is 0.3....
Suppose you toss an unfair coin 8 times independently. The probability ofgetting a head is 0.3. Denote the outcome to be 1 if you get a head and 0 if a tail. (i) Write down the sample space Ω. (ii) What is the probability of the event that you get a head or a tail at least once? (iii) If you get eight same toss's you will get x dollars, otherwise you will lose 1 dollar. On average, how large...
3. (25 pts) A Truly FAIR COIN: Because actual coins are not truly balanced, P -...
3. (25 pts) A Truly FAIR COIN: Because actual coins are not
truly balanced, P - the ACTUAL probability of HEAD for our old,
battered coin - may differ substantially from 1/2. The famous
Mathematician John Von-Neumann came up with the following proposal
for using our possibly unfair coin to simulate a truly fair coin
that always has PROB(HEAD)=PROB(TAIL) = 1/2, as follows: • (i) toss
the UNFAIR coin twice. This is the experiment E. • (ii) IF you got...
5. You play a game using an unfair coin. Suppose that each time the coin is...
5. You play a game using an unfair coin. Suppose that each time the coin is tossed, the probability of showing "head" is 1/3 and the probability of showing "tail" is 2/3. Also suppose that each time the coin shows head you win 10 dollars and you lose 3 dollars when it shows tail. How much money do you expect to win when the coin is tossed 10 times?
A box contains five coins. For each coin there is a different probability that a head...
A box contains five coins. For each coin there is a different probability that a head will be obtained when the coin is tossed. (Some of the coins are not fair coins!) Let pi denote the probability of a head when the i th coin is tossed (i = 1, . . . , 5), and suppose that p1 = 0, p2 =1/4, p3 =1/2, p4 =3/4, p5 =1. The experiment we are interested in consists in selecting at random...
11. What are the possible combination outcomes when you toss a fair coin three times? (6.25...
11. What are the possible combination outcomes when you toss a fair coin three times? (6.25 points) H = Head, T = Tail a {HHH, TTT) Ob. (HHH, TTT, HTH, THT) c. {HHH, TTT, HTH, THT, HHT, TTH, THH) d. (HHH, TTT, HTH, THT, HHT, TTH, THH, HTT} e. None of these 12. What is the probability of you getting three heads straight for tossing a fair coin three times? (6.25 points) a. 1/2 OD. 1/4 C. 118 d. 1/16...
Each game costs $5 and four COINS are flipped simultaneously. If you get one head you get $2, if you get two heads you g...
Each game costs $5 and four COINS are flipped simultaneously. If you get one head you get $2, if you get two heads you get $4, if you get three heads you get $10. Question： create the experimental probability distribution, expected value and bar graph. Compare the distribution, bar graph and expected value to the theoretical. Four Coin Filp ：1-100 Three COINS out of a hundred trials are heads with a probability of 25 Two COINS out of a hundred...
Can you explain how to do parts a-c? 4. Suppose that X is a discrete random...
Can you explain how to do parts a-c?
4. Suppose that X is a discrete random variable with 2 P(X 0) Chapter 8 Estimation of Parameters and Fitting of Probability Distributions P(X = 1) = ) 2 P(X = 3) =-(1-9) where 0 θ 1 is a parameter. The following 10 independent observati were taken from such a distribution: (3, 0, 2, 1, 3, 2, 1, 0, 2, 1). a. Find the method of moments estimate of e. b. Find...
Suppose you have a six sided die. One face is printed with the number 1. Two faces...
Suppose you have a six sided die. One face is printed with the number 1. Two faces are printed with the number 2. Three faces are printed with the number 3. You also have 3 coins: C_1, C_2, and C_3. C_1 will land Heads with probability 1/5. C_2 will land Heads with probability 1/3. C_3 will land Heads with probability 1/2. You roll the die. If the die lands with a 1 face up, flip coin C_1 If the die lands with...
Suppose you are playing two unfair coins. The probability of
tossing a tail is θ for coin 1, and 2θ for coin 2. You toss each
coin several times, and you get the following results:
(a) What is the probability of tossing a head for coin 1 and for
coin 2 [3pts]?
(b) What is the likelihood of the data given θ [7pts]?
(c) What is maximum likelihood estimation for θ [7pts]?
Coin No.Result head head tail head tail tail
3. (25 pts) A Truly FAIR COIN: Because actual coins are not
truly balanced, P - the ACTUAL probability of HEAD for our old,
battered coin - may differ substantially from 1/2. The famous
Mathematician John Von-Neumann came up with the following proposal
for using our possibly unfair coin to simulate a truly fair coin
that always has PROB(HEAD)=PROB(TAIL) = 1/2, as follows: • (i) toss
the UNFAIR coin twice. This is the experiment E. • (ii) IF you got...
5. You play a game using an unfair coin. Suppose that each time the coin is tossed, the probability of showing "head" is 1/3 and the probability of showing "tail" is 2/3. Also suppose that each time the coin shows head you win 10 dollars and you lose 3 dollars when it shows tail. How much money do you expect to win when the coin is tossed 10 times?
11. What are the possible combination outcomes when you toss a fair coin three times? (6.25 points) H = Head, T = Tail a {HHH, TTT) Ob. (HHH, TTT, HTH, THT) c. {HHH, TTT, HTH, THT, HHT, TTH, THH) d. (HHH, TTT, HTH, THT, HHT, TTH, THH, HTT} e. None of these 12. What is the probability of you getting three heads straight for tossing a fair coin three times? (6.25 points) a. 1/2 OD. 1/4 C. 118 d. 1/16...
Can you explain how to do parts a-c?
4. Suppose that X is a discrete random variable with 2 P(X 0) Chapter 8 Estimation of Parameters and Fitting of Probability Distributions P(X = 1) = ) 2 P(X = 3) =-(1-9) where 0 θ 1 is a parameter. The following 10 independent observati were taken from such a distribution: (3, 0, 2, 1, 3, 2, 1, 0, 2, 1). a. Find the method of moments estimate of e. b. Find...
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