Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).
a. X~
b. Graph the probability distribution.
c. f(x)=
d. u=
e. o=
f. Find the probability that a person is born at the exact moment week 19 starts. That is, find P(x=19)=
g. P(2<x|x<28)=
h. Find the probability that a person is born after week 40.
i. P(12<x|x<28)=
j. Find the 70th percentile
k. Find the minimum for the upper quarter.
a. X~Uniform(1, 53)
b. To graph the probability distribution, you can create a bar graph where the x-axis represents the possible values of X (from 1 to 53) and the y-axis represents the probability of each value occurring. Each bar will have a height equal to the probability of that value occurring.
c. The probability density function (PDF) for a uniform distribution can be calculated using the formula: f(x) = 1 / (b-a), where a is the lower limit (1) and b is the upper limit (53). Therefore, f(x) = 1 / 53 for x between 1 and 53. For all other values of x, f(x) = 0.
d. The mean (u) for a uniform distribution can be calculated using the formula: u = (a + b) / 2. In this case, u = (1 + 53) / 2 = 27.
e. The variance (o^2) for a uniform distribution can be calculated using the formula: o^2 = [(b - a + 1)^2 - 1] / 12. In this case, o^2 = [(53 - 1 + 1)^2 - 1] / 12 = 2208 / 12 = 184.
f. To find the probability that a person is born at the exact moment week 19 starts (P(x=19)), we can use the formula: P(x=k) = 1 / (b - a + 1), where k is the specific value (19), a is the lower limit (1), and b is the upper limit (53). Therefore, P(x=19) = 1 / (53 - 1 + 1) = 1 / 53.
g. To find the probability that X is between 2 and 28 (inclusive) (P(2 < x | x < 28)), we can use the formula: P(a < x | x < b) = (b - a) / (53 - 1 + 1), where a is the lower value (2), b is the upper value (28), and 53 is the total range. Therefore, P(2 < x | x < 28) = (28 - 2) / 53 = 26 / 53.
h. To find the probability that a person is born after week 40, we can sum up the probabilities for all weeks beyond week 40. P(X > 40) = P(X = 41) + P(X = 42) + ... + P(X = 53). Since the distribution is uniform, each individual probability is the same (1 / 53). Therefore, P(X > 40) = (53 - 40) / 53 = 13 / 53.
i. To find the probability that X is between 12 and 28 (inclusive) (P(12 < x | x < 28)), we can use the formula mentioned earlier: P(a < x | x < b) = (b - a) / (53 - 1 + 1), where a is the lower value (12), b is the upper value (28), and 53 is the total range. Therefore, P(12 < x | x < 28) = (28 - 12) / 53 = 16 / 53.
j. The 70th percentile represents the value below which 70% of the data falls. To find the 70th percentile, we can calculate the cumulative probability up to that point. Since the probabilities are evenly distributed, we can calculate the value as follows: 70th percentile = a + (b - a) * 0.7, where a is the lower limit (1) and b is the upper limit (53). Therefore, 70th percentile = 1 + (53 - 1) * 0.7 = 1 + 52 * 0.7 = 37.4. Rounded to the nearest whole number, the 70th percentile is 37.
k. The minimum value for the upper quarter represents the lower boundary of the top 25% of the data. To find this value, we can calculate 75% of the total range and add it to the lower limit. Since the distribution is uniform, the upper boundary of the upper quarter is the same as the upper limit (53). Therefore, the minimum value for the upper quarter = (53 - 1) * 0.75 + 1 = 39.25. Rounded to the nearest whole number, the minimum value for the upper quarter is 39.
a. X~ Uniform(1, 53)
b. The probability distribution can be graphed as a horizontal line with height 1/53 for each x between 1 and 53.
c. f(x) = 1/53 for 1 ≤ x ≤ 53, and f(x) = 0 otherwise.
d. The mean (u) of a uniform distribution is given by the formula (a + b) / 2, where a is the lower bound and b is the upper bound. Therefore, u = (1 + 53) / 2 = 27.
e. The variance (o^2) of a uniform distribution is given by the formula (b - a + 1)^2 / 12. Therefore, o^2 = (53 - 1 + 1)^2 / 12 = 52^2 / 12 ≈ 225.33, and o = √225.33 ≈ 15.02.
f. P(x = 19) = 1/53, since each week has an equal probability of 1/53.
g. P(2 < x < 28) = (28 - 2) / 53 = 26 / 53, since there are 27 valid values for x (from 2 to 28) out of a total of 53 possibilities.
h. P(x > 40) = (53 - 40) / 53 = 13 / 53, since there are 13 valid values for x greater than 40 out of a total of 53 possibilities.
i. P(12 < x < 28) = (28 - 12) / 53 = 16 / 53, since there are 17 valid values for x (from 13 to 28) out of a total of 53 possibilities.
j. The 70th percentile is the value that separates the lower 70% of the distribution from the upper 30%. In a uniform distribution, this can be calculated as (b - a + 1) * (percentile/100) + a. Therefore, the 70th percentile is (53 - 1 + 1) * (70/100) + 1 = 53 * 0.7 + 1 = 37.1. Since the result must be a whole number, we can round it down to 37.
k. To find the minimum value for the upper quarter, we need to calculate (b - a + 1) * (3/4) + a. Therefore, the lower bound for the upper quarter is (53 - 1 + 1) * (3/4) + 1 = 53 * 0.75 + 1 = 40.25. Since the result must be a whole number, we can round it up to 41.