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:
(a)
Compute the mean of the Uniformly distributed random variable <em>X</em> as follows:
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:
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)
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:
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:
Thus, the probability that a person is born after week 16 is 0.7115.