Question

a random number generator picks a number from 16 to 57 in a uniform manner. round answers to 4 decimal places when possible.
a. the mean of this distribution is
b. the standard deviation is
c. the probability that the number will be exactly 34 is p(x = 34) =
d. the probability that the number will be between 30 and 49 is p(30 < x < 49) =
e. the probability that the number will be larger than 48 is p(x > 48) =
f. p(x > 28 | x < 35) =
g. find the 37th percentile.
h. find the maximum for the lower quartile.
hint:
written hint
helpful videos: probability +, conditional probability + conditional probability + conditional probability + percentiles +

Explanation:

Step1: Calculate the mean of uniform distribution

For a uniform distribution $X\sim U(a,b)$ where $a = 16$ and $b=57$, the mean $\mu$ is given by $\mu=\frac{a + b}{2}$. So, $\mu=\frac{16+57}{2}=\frac{73}{2}=36.5$.

Step2: Calculate the standard deviation of uniform distribution

The standard - deviation $\sigma$ of a uniform distribution $X\sim U(a,b)$ is $\sigma=\sqrt{\frac{(b - a)^2}{12}}$. Substituting $a = 16$ and $b = 57$, we have $\sigma=\sqrt{\frac{(57 - 16)^2}{12}}=\sqrt{\frac{41^2}{12}}=\sqrt{\frac{1681}{12}}\approx11.8388$.

Step3: Probability of a single value in continuous distribution

In a continuous uniform distribution, the probability of a single value is $0$. So, $P(x = 34)=0$.

Step4: Calculate $P(30\lt x\lt49)$

For a uniform distribution $X\sim U(a,b)$, $P(c\lt x\lt d)=\frac{d - c}{b - a}$. Here, $a = 16$, $b = 57$, $c = 30$, and $d = 49$. So, $P(30\lt x\lt49)=\frac{49 - 30}{57 - 16}=\frac{19}{41}\approx0.4634$.

Step5: Calculate $P(x\gt48)$

$P(x\gt48)=\frac{b - 48}{b - a}$. Substituting $a = 16$ and $b = 57$, we get $P(x\gt48)=\frac{57 - 48}{57 - 16}=\frac{9}{41}\approx0.2195$.

Step6: Calculate $P(x\gt28|x\lt35)$

By the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. For a uniform distribution, $P(x\gt28\cap x\lt35)=\frac{35 - 28}{57 - 16}=\frac{7}{41}$ and $P(x\lt35)=\frac{35 - 16}{57 - 16}=\frac{19}{41}$. So, $P(x\gt28|x\lt35)=\frac{\frac{7}{41}}{\frac{19}{41}}=\frac{7}{19}\approx0.3684$.

Step7: Calculate the 37th percentile

The $k$th percentile of a uniform distribution $X\sim U(a,b)$ is given by $x=a+(b - a)\frac{k}{100}$. For $k = 37$, $a = 16$, and $b = 57$, we have $x=16+(57 - 16)\frac{37}{100}=16 + 41\times0.37=16+15.17 = 31.17$.

Step8: Calculate the lower - quartile

The lower quartile ($25$th percentile) of a uniform distribution $X\sim U(a,b)$ is $x=a+(b - a)\frac{25}{100}$. Substituting $a = 16$ and $b = 57$, we get $x=16+(57 - 16)\times0.25=16 + 41\times0.25=16 + 10.25=26.25$.

Answer:

a. $36.5000$
b. $11.8388$
c. $0.0000$
d. $0.4634$
e. $0.2195$
f. $0.3684$
g. $31.1700$
h. $26.2500$