Question

the time that it takes for the next train to come follows a uniform distribution with f(x) = 1/20 when it goes between 1 and 21 minutes. round answers to 4 decimal places when possible. a. this is a select an answer distribution. b. it is a select an answer distribution. c. find the probability that the time will be at most 6 minutes. d. find the probability that the time will be between 8 and 10 minutes. e. find the 41st percentile. f. find the probability that the time is more than 12 minutes given (or knowing that) it is at least 4 minutes.

Explanation:

Step1: Identify the distribution type

The problem states it's a uniform distribution. For a uniform distribution $X\sim U(a,b)$ where $a = 1$ and $b=21$, and the probability - density function is $f(x)=\frac{1}{b - a}=\frac{1}{21 - 1}=\frac{1}{20}$ for $a\leq x\leq b$.

Step2: Calculate the probability $P(X\leq6)$

The formula for the cumulative - distribution function of a uniform distribution $U(a,b)$ is $F(x)=\frac{x - a}{b - a}$ for $a\leq x\leq b$. Substitute $a = 1$, $b = 21$, and $x = 6$ into the formula: $P(X\leq6)=\frac{6 - 1}{21 - 1}=\frac{5}{20}=0.2500$.

Step3: Calculate the probability $P(8\lt X\lt10)$

Use the formula $P(c\lt X\lt d)=\frac{d - c}{b - a}$ for $a\leq c\lt d\leq b$. Substitute $a = 1$, $b = 21$, $c = 8$, and $d = 10$: $P(8\lt X\lt10)=\frac{10 - 8}{21 - 1}=\frac{2}{20}=0.1000$.

Step4: Calculate the 41st percentile

The formula for the $k$th percentile $x_k$ of a uniform distribution $U(a,b)$ is $x_k=a + k(b - a)/100$. Substitute $a = 1$, $b = 21$, and $k = 41$: $x_{41}=1+\frac{41}{100}(21 - 1)=1 + 41\times\frac{20}{100}=1+8.2 = 9.2000$.

Step5: Calculate the conditional probability $P(X>12|X\geq4)$

By the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A=\{X > 12\}$ and $B=\{X\geq4\}$. $P(A\cap B)=P(X > 12)=\frac{21 - 12}{21 - 1}=\frac{9}{20}$, $P(B)=\frac{21 - 4}{21 - 1}=\frac{17}{20}$. Then $P(X > 12|X\geq4)=\frac{P(X > 12)}{P(X\geq4)}=\frac{\frac{21 - 12}{20}}{\frac{21 - 4}{20}}=\frac{9}{17}\approx0.5294$.

Answer:

c. 0.2500
d. 0.1000
e. 9.2000
f. 0.5294