Continuous Distributions and the Uniform Distribution

Question

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.

a. The mean of this distribution is   .

b. The standard deviation is   .

c. The probability that a person will be born at the exact moment that week 17 begins is

P(x=17)=  .

d. The probability that a person will be born between weeks 14 and 24 is

P (14< x<24)= .

e. The probability that a person will be born after week 21 is

P(x>21)   .

 f. P(x>20 | x<42)= .

g. Find the 53rd percentile. 

h. Find the minimum for the upper quarter.

Answer 1

For the first two, you can use the formulas μ = ^(b-a)/₂ and σ = ^(b-a)/_√12. In this case

(a) mean = (53-1)/2 = 26

(b) s.d. = (53-51)/√12 = 15.0111

c) Because this is a continuous distribution, the probability of being at any exact moment is 0.

d) Probability between 2 times is the difference between those two times and the endpoints: p(14<x<24) = (24-14)/(53-1) = 10/52 = 5/26.

e) After week 21 would be between weeks 21 and 53: P(x>21) = (53-21)/52 = 32/52 = 8/13.

f) For given probability, you're just changing the endpoints in the denominator: P(x>20|x<42) = (42-20)/42-1) = 22/41.

g) Simply solve (x-1)/52 = 0.53. x-1 = 27.56. x = 28.56.

h) Same thing, but it equals 0.75: (x-1)/52 = 0.75. x-1 = 39. x=40.