Coin 1 has bias p1, coin 2 has bias p2, coin 3 has bias p3. All...
Coin 1 has bias p1, coin 2 has bias p2, coin 3 has bias p3. All coin flips are independent.
We choose one of the three coins at random (each coin equally likely). Then we toss n times. Let's say K is A RANDOM VARIABLE the indicates the number of heads. Can we approximate K as normal? If yes what is mean and variance in this case?
Let's say we toss coin 1 n1 times, coin 2 n2 times and coin 3 n3 times. Let's say K is A RANDOM VARIABLE the indicates the number of heads. Can we approximate K as normal? If yes what is mean and variance in this case?
Homework Answers
where here
p1- probability of head when coin 1 is tossed
p2 - probability of head when coin 2 is tossed
p3 - Probability of head when coin 3 is tossed
3. (20 pts.) Jack has three coins C1. C2, and Cs with p. Pp2, and ps as their corresponding p1, p...
3. (20 pts.) Jack has three coins C1. C2, and Cs with p. Pp2, and ps as their corresponding p1, p2, and p3 as their corresponding 3 Wit probabilities of landing heads. Jack flips coin C1 twice and then decides, based on the outcomes, whether to flip coin C2 or C3 next: if the two C1 flips come out the same, Jack flips coin C2 three times; if the two C1 flips come out different, Jack flips coin C3 three...
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A...
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first...
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 t...
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3 if the number of heads is less than 49.5 and B2/3 otherwise. Bound the error probability using the Chernoff bound derived in lecture video (in its final form, after simplifcation).
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3...
2. Mysterioso the Magician is walking down the street with a box containing 25 identical looking coins: 24 are fair coins (which flip heads with probabilty 0.5 and tails with probability 0.5) and one...
2. Mysterioso the Magician is walking down the street with a box containing 25 identical looking coins: 24 are fair coins (which flip heads with probabilty 0.5 and tails with probability 0.5) and one is a trick coin which always flips heads. Renata the Fox skillfully robs Mysterioso of one of the coins in his box (chosen uniformly at random). She decides she will flip the coin k times to test if it is the trick coin (a) What is...
A fair coin is flipped independently until the first Heads is observed. Let the random variable...
A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K = 5. For k 1, 2, , K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xk are independent of one another and of the coin flips. LetX = Σ i Xo Find the...
1. Multiple choice. Circle all the correct answers a) You flip a coin 100,000 times and...
1. Multiple choice. Circle all the correct answers a) You flip a coin 100,000 times and record the outcome in a Xi 1 if the toss is "Heads" and 0 if its "Tails. The Law of Large Numbers says that: i. ii. It is impossible for the first n flips to all be "Heads" if n is large. With high probability, the share of coin flips that are "Heads" will approximate 50%. The sample mean of X is always 0.5...
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...
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...
Consider a pay-to-play game which involves flipping a coin three (3) times. The payout for the...
Consider a pay-to-play game which involves flipping a coin three (3) times. The payout for the game depends on the number of heads obtained in the three coin flips. Let the discrete random variable X represent the number of heads. a) What is the probability P{X = k} associated with each value k of the random variable? b) Suppose that the game has a payout of X^2 dollars. What is the minimum amount that should be charged for admittance (player...
GE 5.52 Find the mean and the standard deviation. If we toss a fair coin 150...
GE 5.52 Find the mean and the standard deviation. If we toss a fair coin 150 times, the number of heads is a random variable that is binomial. (a) Find the mean and the standard deviation of the sample propor- tion of heads. (b) Is your answer to part (a) the same as the mean and the standard deviation of the sample count of heads in 150 throws? Explain your answer
3. (20 pts.) Jack has three coins C1. C2, and Cs with p. Pp2, and ps as their corresponding p1, p2, and p3 as their corresponding 3 Wit probabilities of landing heads. Jack flips coin C1 twice and then decides, based on the outcomes, whether to flip coin C2 or C3 next: if the two C1 flips come out the same, Jack flips coin C2 three times; if the two C1 flips come out different, Jack flips coin C3 three...
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first...
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3 if the number of heads is less than 49.5 and B2/3 otherwise. Bound the error probability using the Chernoff bound derived in lecture video (in its final form, after simplifcation).
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3...
2. Mysterioso the Magician is walking down the street with a box containing 25 identical looking coins: 24 are fair coins (which flip heads with probabilty 0.5 and tails with probability 0.5) and one is a trick coin which always flips heads. Renata the Fox skillfully robs Mysterioso of one of the coins in his box (chosen uniformly at random). She decides she will flip the coin k times to test if it is the trick coin (a) What is...
A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K = 5. For k 1, 2, , K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xk are independent of one another and of the coin flips. LetX = Σ i Xo Find the...
1. Multiple choice. Circle all the correct answers a) You flip a coin 100,000 times and record the outcome in a Xi 1 if the toss is "Heads" and 0 if its "Tails. The Law of Large Numbers says that: i. ii. It is impossible for the first n flips to all be "Heads" if n is large. With high probability, the share of coin flips that are "Heads" will approximate 50%. The sample mean of X is always 0.5...
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...
GE 5.52 Find the mean and the standard deviation. If we toss a fair coin 150 times, the number of heads is a random variable that is binomial. (a) Find the mean and the standard deviation of the sample propor- tion of heads. (b) Is your answer to part (a) the same as the mean and the standard deviation of the sample count of heads in 150 throws? Explain your answer
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