Births are approximately Uniformly distributed between the 52 weeks of the year. They can be said to follow a Uniform distribution from 1 to 53 (a spread of 52 weeks). Round answers to 4 decimal places when possible.
The mean of this distribution is
.
The standard deviation is
.
The probability that a person will be born at the exact moment that week 48 begins is
P(x=48)=
.
The probability that a person will be born between weeks 15 and 47 is
P(15<x<47) =
.
The probability that a person will be born after week 33 is
P(x>33)=
.
P(x>9∣x<18)=
.
Find the 45th percentile.
Find the minimum for the upper quarter
To find the answers to these questions, we can use the properties of the Uniform distribution. In the case of births being uniformly distributed between the 52 weeks of the year, we can apply the following formulas:
1. Mean (μ):
The mean of a Uniform distribution can be calculated using the formula: μ = (a + b) / 2, where a is the lower limit of the distribution (1 in this case) and b is the upper limit (53). So, calculating the mean:
μ = (1 + 53) / 2
μ = 54 / 2
μ = 27
2. Standard Deviation (σ):
The standard deviation of a Uniform distribution can be calculated using the formula: σ = (b - a) / √12. So, in this case:
σ = (53 - 1) / √12
σ = 52 / √12
σ ≈ 15.020
3. Probability of being born at the exact moment of week 48 (P(x = 48)):
Since births are uniformly distributed, the probability of being born in any given week is equal. So, the probability of being born in week 48 or any other specific week is 1 / 52. Therefore:
P(x = 48) = 1 / 52 ≈ 0.0192
4. Probability of being born between weeks 15 and 47 (P(15 < x < 47)):
The probability of an event occurring within a range in a Uniform distribution can be calculated by taking the difference between the upper and lower bounds and dividing it by the total range. So, in this case:
P(15 < x < 47) = (47 - 15) / (53 - 1) = 32 / 52 ≈ 0.6154
5. Probability of being born after week 33 (P(x > 33)):
To find the probability of being born after a specific week, we can subtract the cumulative probability of being born on or before that week from 1. So, in this case:
P(x > 33) = 1 - (33 / 52) ≈ 0.3654
6. Conditional Probability of being born after week 9 given that the person is born before week 18 (P(x > 9 | x < 18)):
Conditional probability can be calculated by dividing the probability of the joint event by the probability of the condition. In this case:
P(x > 9 | x < 18) = P(10 ≤ x ≤ 17) / P(x < 18)
P(x > 9 | x < 18) = (17 - 10 + 1) / 52 / (18 - 1 + 1) / 52 = 8 / 18 ≈ 0.4444
7. 45th Percentile:
The percentile can be calculated by multiplying the total range by the desired percentile and adding the lower limit. In this case, the 45th percentile would be:
45th Percentile = (52 * 0.45) + 1 ≈ 24.4
8. Minimum for the upper quarter:
The upper quarter refers to the top 25% of the distribution. Since it is a uniform distribution, the minimum value in the upper quarter would be the upper limit (b) minus one-fourth of the range. In this case:
Minimum for the upper quarter = 53 - (52 / 4) = 39.5
So, to summarize:
Mean (μ) ≈ 27
Standard Deviation (σ) ≈ 15.020
P(x = 48) ≈ 0.0192
P(15 < x < 47) ≈ 0.6154
P(x > 33) ≈ 0.3654
P(x > 9 | x < 18) ≈ 0.4444
45th Percentile ≈ 24.4
Minimum for upper quarter ≈ 39.5