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]?
Homework Answers
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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...
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...
2 Maximum Likelihood [25pts] 2.1 Discrete Example 17pts) Suppose you are playing two unfair coins. The...
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...
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...
An experiment consists of tossing an unfair coin (53% chance of landing on heads) a specified...
An experiment consists of tossing an unfair coin (53% chance of landing on heads) a specified number of times and recording the outcomes. (a) What is the probability that the first head will occur on the second trial? (Use 4 decimal places.) Does this probability change if we toss the coin three times? What if we toss the coin four times? The probability changes if we toss the coin three times, but does not change if we toss the coin...
An experiment consists of tossing an unfair coin (49% chance of landing on heads) a specified...
An experiment consists of tossing an unfair coin (49% chance of landing on heads) a specified number of times and recording the outcomes. (a) What is the probability that the first head will occur on the second trial? (Use 4 decimal places.) Does this probability change if we toss the coin three times? What if we toss the coin four times? The probability changes if we toss the coin four times, but does not change if we toss the coin...
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...
A fair coin is tossed 6 times. A) What is the probability of tossing a tail...
A fair coin is tossed 6 times. A) What is the probability of tossing a tail on the 6th toss given the preceding 5 tosses were heads? B) What is the probability of getting either 6 heads or 6 tails?
3 Suppose that a box contains five coins, and that for each coin there is a different probability...
3 Suppose that a box contains five coins, and that for each coin there is a different probability that a head will be obtained when the coin is tossed. Let pi denote the probability of a head when the ith coin is tossed, where i 1,2,3, 4,5]. Suppose that a (8 marks) Suppose that one coin is selected at random from the the probability that the ith coin was selected? Note that i b (8 marks) If the same coin...
We are given 3 coins. The first coin, coin X, has a head on both sides,...
We are given 3 coins. The first coin, coin X, has a head on both sides, the second coin, coin Y, has a head on one side and a tail on the other and the third coin, coin Z, has a tail on both sides. You pick a coin among the three coins at random and with equal likelihood of picking any one of the three coins X,Y,Z. You then toss the coin and a tail shows up. What is...
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?
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...
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...
3 Suppose that a box contains five coins, and that for each coin there is a different probability that a head will be obtained when the coin is tossed. Let pi denote the probability of a head when the ith coin is tossed, where i 1,2,3, 4,5]. Suppose that a (8 marks) Suppose that one coin is selected at random from the the probability that the ith coin was selected? Note that i b (8 marks) If the same coin...
We are given 3 coins. The first coin, coin X, has a head on both sides, the second coin, coin Y, has a head on one side and a tail on the other and the third coin, coin Z, has a tail on both sides. You pick a coin among the three coins at random and with equal likelihood of picking any one of the three coins X,Y,Z. You then toss the coin and a tail shows up. What 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?
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