The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 11 seconds. (a) Sketch this exponential probability distribution. (b) What is the probability that the arrival time between vehicles is 11 seconds or less? (c) What is the probability that the arrival time between vehicles is 7 seconds or less? (d) What is the probability of 33 or more seconds between vehicle arrivals?

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.3: Solving Linear Equations
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The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 11 seconds.
(a)
Sketch this exponential probability distribution.
(b)
What is the probability that the arrival time between vehicles is 11 seconds or less?
(c)
What is the probability that the arrival time between vehicles is 7 seconds or less?
(d)
What is the probability of 33 or more seconds between vehicle arrivals?
Step 1

(a)Sketch this exponential probability distribution.

The exponential probability distribution is below where μ is the expected value, or mean, and 
x ≥ 0.
f(x) = 
1
μ
e−x/μ
It is given that vehicles arrive at an intersection with a mean of 11 seconds, so 
μ =  .
 Substitute the value of μ into the probability distribution above and graph the resulting curve.
A graph with 1 continuous curve is given. The area under the curve is shaded.
  • The horizontal axis is labeled: x and ranges from 0 to 30.
  • The vertical axis is labeled: f(x) and ranges from 0 to 0.15.
  • The curve begins near (0, 0.13) and curves down to the right.
  • It leaves the viewing window near (30, 0).
 
A graph with 1 continuous curve is given. The area under the curve is shaded.
  • The horizontal axis is labeled: x and ranges from 0 to 30.
  • The vertical axis is labeled: f(x) and ranges from 0 to 0.15.
  • The curve begins near (0, 0.09) and curves down to the right.
  • It leaves the viewing window near (30, 0.01).
 
A graph with 1 continuous curve is given. The area under the curve is shaded.
  • The horizontal axis is labeled: x and ranges from 0 to 30.
  • The vertical axis is labeled: f(x) and ranges from 0 to 0.15.
  • The curve begins near (0, 0.07) and curves down to the right.
  • It leaves the viewing window near (30, 0.01).
 
A graph with 1 continuous curve is given. The area under the curve is shaded.
  • The horizontal axis is labeled: x and ranges from 0 to 30.
  • The vertical axis is labeled: f(x) and ranges from 0 to 0.15.
  • The curve begins near (0, 0.06) and curves down to the right.
  • It leaves the viewing window near (30, 0.01).
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