Answer:
(a) To find the variance and standard deviation of the number of bankruptcies filed per hour by businesses in a country, we can use the Poisson distribution. The mean number of bankruptcies filed per hour is 8.
The variance of a Poisson distribution is equal to its mean. Therefore, the variance of the number of bankruptcies filed per hour is also 8.
The standard deviation of a Poisson distribution is the square root of its variance. Therefore, the standard deviation of the number of bankruptcies filed per hour is approximately 2.83 .
Interpretation: The variance of 8 indicates that the number of bankruptcies filed per hour by businesses in a country is not very spread out. The standard deviation of 2.83 indicates that the number of bankruptcies filed per hour by businesses in a country is relatively close to the mean of 8.
(b) To find the probability that at most three businesses will file bankruptcy in any given hour, we can use the Poisson distribution. The mean number of bankruptcies filed per hour is 8.
The probability that at most three businesses will file bankruptcy in any given hour is the sum of the probabilities that 0, 1, 2, or 3 businesses will file bankruptcy in any given hour. Using the Poisson distribution, we can calculate this probability as follows:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
where X is the number of bankruptcies filed per hour.
Using the Poisson distribution formula, we can calculate each of these probabilities as follows:
P(X = k) = (e^-λ * λ^k) / k!
λ is the mean number of bankruptcies filed per hour and k is the number of bankruptcies filed per hour.
Therefore, we have:
P(X = 0) = (e^-8 * 8^0) / 0! ≈ 0.000335
P(X = 1) = (e^-8 * 8^1) / 1! ≈ 0.002687
P(X = 2) = (e^-8 * 8^2) / 2! ≈ 0.010747
P(X = 3) = (e^-8 * 8^3) / 3! ≈ 0.028658
The probability that at most three businesses will file bankruptcy in any given hour is:
P(X ≤ 3) ≈ 0.042427
The probability that at most three businesses will file bankruptcy in any given hour is approximately 0.042427 or 4.24%.
Explanation:
^^