Question

1. at the george bush intercontinental airport in houston, texas, the skyway automated people - mover is an above - ground train that shuttles travelers from one terminal to another. it departs every two minutes from the skyway stations. let ( x=) time spent waiting (in seconds) for the train to depart for a randomly selected passenger. ( x ) is a uniform random variable.

a. use the axes to create the graph of the probability distribution of ( x ).

time spent waiting (seconds)
b. find the probability that a randomly selected passenger waits less than 45 seconds.
c. find ( p(x > 90 \text{ seconds}) ).
d. find ( p(30leq xleq85) ).

Explanation:

Step1: Define uniform distribution bounds

The train departs every 2 minutes (120 seconds), so $X \sim \text{Uniform}(a=0, b=120)$. The probability density function (pdf) is $f(x) = \frac{1}{b-a} = \frac{1}{120}$ for $0 \leq x \leq 120$, and 0 otherwise.

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Part a: Graph the distribution

Step1: Set up axes

x-axis: "Time spent waiting (seconds)" from 0 to 120.
y-axis: Probability density, scaled to $\frac{1}{120} \approx 0.0083$.

Step2: Draw the pdf

Draw a horizontal line at $y = \frac{1}{120}$ from $x=0$ to $x=120$, with vertical segments at $x=0$ and $x=120$ connecting to the x-axis.

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Part b: Calculate $P(X < 45)$

Step1: Apply uniform probability formula

For uniform distribution, $P(X < k) = \frac{k - a}{b - a}$

Step2: Substitute values

$P(X < 45) = \frac{45 - 0}{120 - 0} = \frac{45}{120} = \frac{3}{8}$

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Part c: Calculate $P(X > 90)$

Step1: Apply upper tail probability formula

$P(X > k) = \frac{b - k}{b - a}$

Step2: Substitute values

$P(X > 90) = \frac{120 - 90}{120 - 0} = \frac{30}{120} = \frac{1}{4}$

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Part d: Calculate $P(30 \leq X \leq 85)$

Step1: Apply interval probability formula

$P(c \leq X \leq d) = \frac{d - c}{b - a}$

Step2: Substitute values

$P(30 \leq X \leq 85) = \frac{85 - 30}{120 - 0} = \frac{55}{120} = \frac{11}{24}$

Answer:

a. A horizontal line at $y=\frac{1}{120}$ spanning $x=0$ to $x=120$, with vertical endpoints at (0,0) to (0, 1/120) and (120,0) to (120, 1/120).
b. $\frac{3}{8}$
c. $\frac{1}{4}$
d. $\frac{11}{24}$