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 born at the exact moment that week 36 begins is Px 36) between weeks 8 and 45 is P(8 < ®?45)- The probability that a person will be born after week 16 is P(x > b. The standard deviation is c. The probability that a person will be d. The probability that a person will be born h. Find the minimum for the lower f. P(x>201 x 46). Find the 48th percentile quartile.

Answer 1

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:

$f_{X}(x)=\left \{ {{\frac{1}{b-a};\ a$

(a)

Compute the mean of the Uniformly distributed random  variable X as follows:

$E(X)=\frac{1}{2}(a+b)$

         $=\frac{1}{2}\times (1+53)$

         $=27$

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:

$SD(X)=\sqrt{\frac{1}{12}\times (b-a)^{2}}$

            $=\sqrt{\frac{1}{12}\times (53-1)^{2}}$

            $=15.01$

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)

                $=\int\limits^{36.5}_{35.5}{\frac{1}{53-1}}}\, dx$

                $=\frac{1}{52}\times \int\limits^{36.5}_{35.5}{1}\, dx$

                $=\frac{36.5-35.5}{52}$

                $=0.0192$

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:

$P(8$

                        $=\frac{1}{52}\times \int\limits^{45}_{8}{1}\, dx$

                        $=\frac{45-8}{52}$

                        $=0.7115$

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:

$P(X>16)=\int\limits^{53}_{16}{\frac{1}{53-1}}\, dx$

                 $=\frac{1}{52}\times \int\limits^{53}_{16}{1}\, dx$

                 $=\frac{53-16}{52}$

                 $=0.7115$

Thus, the probability that a person is born after week 16 is 0.7115.