Question

6.31 Waiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for the train is a variable with density curve $y=1/30$for $0<x<30$, and $y=0$ otherwise.
a. Graph the density curve of this variable.
b. Show that the area under this density curve to the left of any number $x$between 0 and 30 equals $x/30$.

What percentage of the time does John wait for the train
c. less than 5 minutes?
d. between 10 and 15 minutes?
e. at least 20 minutes?

Short answer

(a) The density function graph can be drawn as:

(b) Shown that the area under this density curve to the left of any number $x$between 0 and 30 equals $x/30$.

(c) The percentage of the time does John wait for the train less than 5 minutes is $16.67\%$.

(d) The percentage of the time does John wait for the train less than 10 and 15 minutes is $16.67\%$.

(e) The percentage of the time for which the John waits for the train is at least 20 minutes is $33.33\%$.

Step-by-step solution

Part (a) Step 1: Given information

To graph the density curve of this variable.

Part (a) Step 2: Explanation

The random variable $x$represents the amount of time John waits for the train in minutes.$y=\left\{\begin{array}{l}\frac{1}{30}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{for}0<x<30\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{otherwise}\end{array}\right.$
The density function graph can be drawn as:

Part (b) Step 1: Given information

To show that the area under this density curve to the left of any number $x$between 0 and 30 equals $x/30$.

Part (b) Step 2: Explanation

The base of the rectangle is $x$, and the height of the rectangle is $1/30$, as seen above.
As a result, the rectangle's area is
$\text{Area}=\text{Base}\times \text{Height}$
$=\left(x\right)\left(\frac{1}{30}\right)$
$=\frac{x}{30}$
Since, it is proved

Part (c) Step 1: Given information

To find the percentage of the time does John wait for the train less than 5 minutes.

Part (c) Step 2: Explanation

Determine the percentage of time spent waiting for the train by John is less than 5 minutes, as follows:
Area less than 5 minutes as:

$=\frac{5}{30}$

$=0.1667$

$=16.67\%$

As a result, the percentage of the time does John wait for the train less than 5 minutes is $16.67\%$.

Part (d) Step 1: Given information

To find the percentage of the time does John wait for the train between 10 and 15 minutes.

Part (d) Step 2: Explanation

Determine the percentage of the time for which the John waits for the train is between 10 and 15 minutes as follows:
Area between 10 and 15= Area to the left of 15 - Area to the left of 10
$=\frac{15}{30}-\frac{10}{30}$
$=\frac{5}{30}$
$=16.67\%$
As a result, the percentage of the time for which the John waits for the train is between 10 and 15 minutes is $16.67\%$.

Part (e) Step 1: Given information

To determine the percentage of the time for which the John waits for the train is at least 20 minutes.

Part (e) Step 2: Explanation

Determine the percentage of the time for which the John waits for the train is at least 20 minutes as follows:
Area to the right of 20 = 1 - Area to the left of 20
$=1-\frac{20}{30}$
$=\frac{10}{30}$
$=33.33\%$

Asa result, the percentage of the time for which the John waits for the train is at least 20 minutes is $33.33\%$.