Question

question 4
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 42 begins is p(x = 42) =
d. the probability that a person will be born between weeks 11 and 38 is p(11 < x < 38) =
e. the probability that a person will be born after week 31 is p(x > 31) =
f. p(x > 18 | x < 35) =
g. find the 28th percentile.
h. find the minimum for the upper quartile.

Explanation:

Step1: Recall mean formula for uniform distribution

For a uniform distribution $U(a,b)$, the mean $\mu=\frac{a + b}{2}$. Here $a = 1$ and $b=52$, so $\mu=\frac{1+52}{2}=26.5$.

Step2: Recall standard - deviation formula for uniform distribution

The standard deviation $\sigma=\sqrt{\frac{(b - a)^2}{12}}$. Substituting $a = 1$ and $b = 52$, we have $\sigma=\sqrt{\frac{(52 - 1)^2}{12}}=\sqrt{\frac{51^2}{12}}=\sqrt{\frac{2601}{12}}\approx14.9145$.

Step3: Probability at a single point for continuous distribution

For a continuous uniform distribution, the probability at a single point is $0$. So $P(x = 42)=0$.

Step4: Calculate $P(11\lt x\lt38)$

The probability density function of a uniform distribution $U(a,b)$ is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$. Here $f(x)=\frac{1}{52 - 1}=\frac{1}{51}$. Then $P(11\lt x\lt38)=\frac{38 - 11}{52 - 1}=\frac{27}{51}\approx0.5294$.

Step5: Calculate $P(x\gt31)$

$P(x\gt31)=\frac{52 - 31}{52 - 1}=\frac{21}{51}\approx0.4118$.

Step6: Calculate $P(x\gt18|x\lt35)$

Using the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here $A=\{x\gt18\}$ and $B = \{x\lt35\}$, so $A\cap B=\{18\lt x\lt35\}$. Then $P(x\gt18|x\lt35)=\frac{35 - 18}{35 - 1}=\frac{17}{34}=0.5$.

Step7: Find the 28th percentile

Let the 28th percentile be $k$. We use the formula $P(x\lt k)=0.28$. Since $P(x\lt k)=\frac{k - 1}{52 - 1}$, then $\frac{k - 1}{51}=0.28$, and $k=1+0.28\times51=1 + 14.28 = 15.28$.

Step8: Find the minimum for the upper quartile

The upper quartile is the 75th percentile. Let it be $q$. Using the formula $P(x\lt q)=\frac{q - 1}{52 - 1}=0.75$. Then $q=1+0.75\times51=1 + 38.25 = 39.25$.

Answer:

a. $26.5000$
b. $14.9145$
c. $0.0000$
d. $0.5294$
e. $0.4118$
f. $0.5000$
g. $15.2800$
h. $39.2500$