Respuesta :
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 X can be defined as the births between the 52 weeks of the year.
The random variable X is uniformly distributed with parameters a = 1 to b = 53.
The probability distribution function of X is:
[tex]f_{X}(x)=\left \{ {{\frac{1}{b-a};\ a<X<b;\ a<b} \atop {0;\ otherwise}} \right.[/tex]
(a)
Compute the mean of the Uniformly distributed random variable X as follows:
[tex]E(X)=\frac{1}{2}(a+b)[/tex]
[tex]=\frac{1}{2}\times (1+53)[/tex]
[tex]=27[/tex]
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 X as follows:
[tex]SD(X)=\sqrt{\frac{1}{12}\times (b-a)^{2}}[/tex]
[tex]=\sqrt{\frac{1}{12}\times (53-1)^{2}}[/tex]
[tex]=15.01[/tex]
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)
[tex]=\int\limits^{36.5}_{35.5}{\frac{1}{53-1}}}\, dx\\[/tex]
[tex]=\frac{1}{52}\times \int\limits^{36.5}_{35.5}{1}\, dx[/tex]
[tex]=\frac{36.5-35.5}{52}[/tex]
[tex]=0.0192[/tex]
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:
[tex]P(8<X<45)=\int\limits^{45}_{8}{\frac{1}{53-1}}\, dx[/tex]
[tex]=\frac{1}{52}\times \int\limits^{45}_{8}{1}\, dx[/tex]
[tex]=\frac{45-8}{52}[/tex]
[tex]=0.7115[/tex]
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:
[tex]P(X>16)=\int\limits^{53}_{16}{\frac{1}{53-1}}\, dx[/tex]
[tex]=\frac{1}{52}\times \int\limits^{53}_{16}{1}\, dx[/tex]
[tex]=\frac{53-16}{52}[/tex]
[tex]=0.7115[/tex]
Thus, the probability that a person is born after week 16 is 0.7115.
