Tossing a triple of coins
We have a red coin, for which P(Heads)=0.4, a green coin, for which P(Heads)=0.5, and a yellow coin, for which P(Heads)=0.6. The flips of the same or of different coins are independent. For each of the following situations, determine whether the random variable N can be approximated by a normal.
If yes, enter the mean and variance of N. If not, enter 0 in both of the corresponding answer boxes.
1) Let N be the number of Heads in 300 tosses of the red coin.
mean:
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Variance:
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2) Let N be the number of Heads in 300 tosses. At each toss, one of the three coins is selected at random (either choice is equally likely), and independently from everything else.
mean:
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variance:
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3) Let N be the number of Heads in 100 tosses of the red coin, followed by 100 tosses of the green coin, followed by 100 tosses of the yellow coin (for a total of 300 tosses).
mean:
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variance:
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4) We select one of the three coins at random: each coin is equally likely to be selected. We then toss the selected coin 300 times, independently, and let N be the number of Heads.
mean:
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variance:
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To determine whether the random variable N can be approximated by a normal distribution, we need to check if the conditions for the Central Limit Theorem are satisfied. The Central Limit Theorem states that if we have a large enough sample size and the random variable is independently and identically distributed, then the distribution of the sample mean will be approximately normal.
1) Let N be the number of Heads in 300 tosses of the red coin.
Since the random variable N represents the number of Heads in 300 tosses of the red coin, the sample size is large enough (300). However, the random variable is not identically distributed as the probability of Heads for the red coin is 0.4, which is different from 0.5. Therefore, N cannot be approximated by a normal distribution.
mean: 0
variance: 0
2) Let N be the number of Heads in 300 tosses. At each toss, one of the three coins is selected at random (either choice is equally likely), and independently from everything else.
In this case, the random variable N represents the number of Heads in 300 tosses, where at each toss, one of the three coins is selected at random. The sample size is large enough (300), and each toss is independent. However, the coin selection is not identically distributed as each coin has a different probability of Heads. Therefore, N cannot be approximated by a normal distribution.
mean: 0
variance: 0
3) Let N be the number of Heads in 100 tosses of the red coin, followed by 100 tosses of the green coin, followed by 100 tosses of the yellow coin (for a total of 300 tosses).
In this case, the random variable N represents the number of Heads in 100 tosses of the red coin, followed by 100 tosses of the green coin, followed by 100 tosses of the yellow coin. The sample size is large enough (300), and each toss within each group is independent. Since each group has a different coin with a different probability of Heads, the random variable N cannot be approximated by a normal distribution.
mean: 0
variance: 0
4) We select one of the three coins at random: each coin is equally likely to be selected. We then toss the selected coin 300 times, independently, and let N be the number of Heads.
In this case, the random variable N represents the number of Heads in 300 tosses of one of the three coins, selected at random. The sample size is large enough (300), and each toss of the selected coin is independent. Since the coin selection is random and each coin has a different probability of Heads, the random variable N cannot be approximated by a normal distribution.
mean: 0
variance: 0
To determine whether the random variable N can be approximated by a normal distribution, we need to check if certain conditions are met. The conditions are based on the properties of the normal distribution and the central limit theorem.
The conditions for approximation by a normal distribution are:
1) The random variable N should be the sum of a large number of independent and identically distributed (i.i.d.) random variables.
2) The underlying probability distribution of each individual random variable should not be too skewed or have extreme outliers.
Now, let's analyze each situation to determine whether the conditions are met and find the mean and variance if they are.
1) Situation: N is the number of Heads in 300 tosses of the red coin.
In this case, N is the sum of 300 independent and identically distributed Bernoulli random variables with probability of success (P(Heads)) of 0.4. The conditions for approximation by a normal distribution are met because the number of tosses is large (300), and the individual probabilities (0.4) are not extreme.
To find the mean and variance of N, we can use the properties of the Bernoulli distribution:
Mean = n * P(Heads) = 300 * 0.4 = 120
Variance = n * P(Heads) * (1 - P(Heads)) = 300 * 0.4 * (1 - 0.4) = 72
Mean: 120
Variance: 72
2) Situation: N is the number of Heads in 300 tosses, with a random selection of one of the three coins at each toss.
In this case, N is the sum of 300 independent and identically distributed random variables, where the probability distribution changes at each toss depending on the randomly selected coin. The conditions for approximation by a normal distribution are not met because the probabilities of success (Heads) change at each toss.
Mean: 0
Variance: 0
3) Situation: N is the number of Heads in 100 tosses each of the red, green, and yellow coins.
In this case, N is the sum of 300 independent and identically distributed Bernoulli random variables, but with different probabilities of success for each coin. The conditions for approximation by a normal distribution are met because the number of tosses is large (100 for each coin), and the probabilities are within a reasonable range.
To find the mean and variance of N, we can sum up the means and variances of each coin's tosses:
Mean = n * P(Heads) = (100 * 0.4) + (100 * 0.5) + (100 * 0.6) = 40 + 50 + 60 = 150
Variance = n * P(Heads) * (1 - P(Heads)) = (100 * 0.4 * (1 - 0.4)) + (100 * 0.5 * (1 - 0.5)) + (100 * 0.6 * (1 - 0.6)) = 24 + 25 + 24 = 73
Mean: 150
Variance: 73
4) Situation: N is the number of Heads in 300 tosses of a randomly selected coin.
In this case, N is the sum of 300 independent and identically distributed random variables, where the probability distribution changes based on the randomly selected coin. The conditions for approximation by a normal distribution are not met because the probabilities of success (Heads) change.
Mean: 0
Variance: 0
In summary, the answers are as follows:
1) Mean: 120, Variance: 72
2) Mean: 0, Variance: 0
3) Mean: 150, Variance: 73
4) Mean: 0, Variance: 0
1) No, the random variable N cannot be approximated by a normal distribution, since the number of heads is based on a single coin with a fixed probability.
2) Yes, the random variable N can be approximated by a normal distribution. The mean would be (300/3) * 0.5 = 50, and the variance would be (300/3) * 0.5 * (1 - 0.5) = 25.
3) No, the random variable N cannot be approximated by a normal distribution, since the number of heads is based on different coins with different probabilities.
4) Yes, the random variable N can be approximated by a normal distribution. The mean would be (300/3) * ((0.4 + 0.5 + 0.6) / 3) = 50, and the variance would be (300/3) * (((0.4 - 0.5)² + (0.5 - 0.5)² + (0.6 - 0.5)²) / 3) = 8.33.