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