Answer:
(a) The mean of the distribution of births between the 52 weeks of a year is 27.
(b) The standard deviation of the distribution of births between the 52 weeks of a year is 15.01.
(c) The probability that a person is born exactly at the beginning of week 36 is 0.0192.
(d) The probability that a person will be born between weeks 8 and 45 is 0.7115.
(e) The probability that a person is born after week 16 is 0.7115.
Step-by-step explanation:
The random variable <em>X</em> can be defined as the births between the 52 weeks of the year.
The random variable <em>X</em> is uniformly distributed with parameters <em>a</em> = 1 to <em>b</em> = 53.
The probability distribution function of <em>X</em> is:
![f_{X}(x)=\left \{ {{\frac{1}{b-a};\ a](https://tex.z-dn.net/?f=f_%7BX%7D%28x%29%3D%5Cleft%20%5C%7B%20%7B%7B%5Cfrac%7B1%7D%7Bb-a%7D%3B%5C%20a%3CX%3Cb%3B%5C%20a%3Cb%7D%20%5Catop%20%7B0%3B%5C%20otherwise%7D%7D%20%5Cright.)
(a)
Compute the mean of the Uniformly distributed random variable <em>X</em> as follows:
![E(X)=\frac{1}{2}(a+b)](https://tex.z-dn.net/?f=E%28X%29%3D%5Cfrac%7B1%7D%7B2%7D%28a%2Bb%29)
![=\frac{1}{2}\times (1+53)](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20%281%2B53%29)
![=27](https://tex.z-dn.net/?f=%3D27)
Thus, the mean of the distribution of births between the 52 weeks of a year is 27.
(b)
Compute the standard deviation of the Uniformly distributed random variable <em>X</em> as follows:
![SD(X)=\sqrt{\frac{1}{12}\times (b-a)^{2}}](https://tex.z-dn.net/?f=SD%28X%29%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B12%7D%5Ctimes%20%28b-a%29%5E%7B2%7D%7D)
![=\sqrt{\frac{1}{12}\times (53-1)^{2}}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B12%7D%5Ctimes%20%2853-1%29%5E%7B2%7D%7D)
![=15.01](https://tex.z-dn.net/?f=%3D15.01)
Thus, the standard deviation of the distribution of births between the 52 weeks of a year is 15.01.
(c)
Compute the probability that a person is born exactly at the beginning of week 36 as follows:
Use continuity correction.
P (X = 36) = P (36 - 0.5 < X < 36 + 0.5)
= P (35.5 < X < 36.5)
![=\int\limits^{36.5}_{35.5}{\frac{1}{53-1}}}\, dx\\](https://tex.z-dn.net/?f=%3D%5Cint%5Climits%5E%7B36.5%7D_%7B35.5%7D%7B%5Cfrac%7B1%7D%7B53-1%7D%7D%7D%5C%2C%20dx%5C%5C)
![=\frac{1}{52}\times \int\limits^{36.5}_{35.5}{1}\, dx](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B52%7D%5Ctimes%20%5Cint%5Climits%5E%7B36.5%7D_%7B35.5%7D%7B1%7D%5C%2C%20dx)
![=\frac{36.5-35.5}{52}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B36.5-35.5%7D%7B52%7D)
![=0.0192](https://tex.z-dn.net/?f=%3D0.0192)
Thus, the probability that a person is born exactly at the beginning of week 36 is 0.0192.
(d)
Compute the probability that a person will be born between weeks 8 and 45 as follows:
![P(8](https://tex.z-dn.net/?f=P%288%3CX%3C45%29%3D%5Cint%5Climits%5E%7B45%7D_%7B8%7D%7B%5Cfrac%7B1%7D%7B53-1%7D%7D%5C%2C%20dx)
![=\frac{1}{52}\times \int\limits^{45}_{8}{1}\, dx](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B52%7D%5Ctimes%20%5Cint%5Climits%5E%7B45%7D_%7B8%7D%7B1%7D%5C%2C%20dx)
![=\frac{45-8}{52}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B45-8%7D%7B52%7D)
![=0.7115](https://tex.z-dn.net/?f=%3D0.7115)
Thus, the probability that a person will be born between weeks 8 and 45 is 0.7115.
(e)
Compute the probability that a person is born after week 16 as follows:
![P(X>16)=\int\limits^{53}_{16}{\frac{1}{53-1}}\, dx](https://tex.z-dn.net/?f=P%28X%3E16%29%3D%5Cint%5Climits%5E%7B53%7D_%7B16%7D%7B%5Cfrac%7B1%7D%7B53-1%7D%7D%5C%2C%20dx)
![=\frac{1}{52}\times \int\limits^{53}_{16}{1}\, dx](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B52%7D%5Ctimes%20%5Cint%5Climits%5E%7B53%7D_%7B16%7D%7B1%7D%5C%2C%20dx)
![=\frac{53-16}{52}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B53-16%7D%7B52%7D)
![=0.7115](https://tex.z-dn.net/?f=%3D0.7115)
Thus, the probability that a person is born after week 16 is 0.7115.