The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1 βe−x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. If a and b are any numbers such that 0 < a < b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is the following. P(a < x < b) = e−a/β − e−b/β Notice that by definition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are the following. μ = β and σ = β Note: The number e = 2.71828 is used throughout probability, statistics, and mathematics. The key ex is conveniently located on most calculators. Comment: The Poisson and exponential distributions have a special relationship. Specifically, it can be shown that the waiting time between successive Poisson arrivals (i.e., successes or rare events) has an exponential distribution with β = 1/λ, where λ is the average number of Poisson successes (rare events) per unit of time. Fatal accidents on scheduled domestic passenger flights are rare events. In fact, airlines do all they possibly can to prevent such accidents. However, around the world such fatal accidents do occur. Let x be a random variable representing the waiting time between fatal airline accidents. Research has shown that x has an exponential distribution with a mean of approximately 44 days.† We take the point of view that x (measured in days as units) is a continuous random variable. Suppose a fatal airline accident has just been reported on the news. What is the probability that the waiting time to the next reported fatal airline accident is the following?

The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1 βe−x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. If a and b are any numbers such that 0 < a < b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is the following. P(a < x < b) = e−a/β − e−b/β Notice that by definition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are the following. μ = β and σ = β Note: The number e = 2.71828 is used throughout probability, statistics, and mathematics. The key ex is conveniently located on most calculators. Comment: The Poisson and exponential distributions have a special relationship. Specifically, it can be shown that the waiting time between successive Poisson arrivals (i.e., successes or rare events) has an exponential distribution with β = 1/λ, where λ is the average number of Poisson successes (rare events) per unit of time. Fatal accidents on scheduled domestic passenger flights are rare events. In fact, airlines do all they possibly can to prevent such accidents. However, around the world such fatal accidents do occur. Let x be a random variable representing the waiting time between fatal airline accidents. Research has shown that x has an exponential distribution with a mean of approximately 44 days.† We take the point of view that x (measured in days as units) is a continuous random variable. Suppose a fatal airline accident has just been reported on the news. What is the probability that the waiting time to the next reported fatal airline accident is the following?

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter14: Counting And Probability

Section14.2: Probability

Problem 3E: The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the...

See similar textbooks

Related questions

Question

The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let

x > 0

be a random variable and let

β > 0

be a constant. Then

y =

e^−x/β

is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities.

If a and b are any numbers such that

0 < a < b,

then using some extra mathematics, it can be shown that the area under the curve above the interval

[a, b]

is the following.

P(a < x < b) = e^−a/β − e^−b/β

Notice that by definition, x cannot be negative, so,

P(x < 0) = 0.

The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are the following.

μ = β and σ = β

Note: The number

e = 2.71828

is used throughout probability, statistics, and mathematics. The key

e^x

is conveniently located on most calculators.

Comment: The Poisson and exponential distributions have a special relationship. Specifically, it can be shown that the waiting time between successive Poisson arrivals (i.e., successes or rare events) has an exponential distribution with

β = 1/λ,

where λ is the average number of Poisson successes (rare events) per unit of time.

Fatal accidents on scheduled domestic passenger flights are rare events. In fact, airlines do all they possibly can to prevent such accidents. However, around the world such fatal accidents do occur. Let x be a random variable representing the waiting time between fatal airline accidents. Research has shown that x has an exponential distribution with a mean of approximately 44 days.†

We take the point of view that x (measured in days as units) is a continuous random variable. Suppose a fatal airline accident has just been reported on the news. What is the probability that the waiting time to the next reported fatal airline accident is the following?

(a) less than 30 days

(i.e., 0 ≤ x < 30)

(Round your answer to four decimal places.)

(b) more than 60 days

(i.e., 60 < x < ∞)

Hint:

e^−∞ = 0

(Round your answer to four decimal places.)

(c) between 20 and 50 days (Round your answer to four decimal places.)

(d) What is the mean of the waiting times x?
days

What is the standard deviation of the waiting times x?
days

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Multivariate Distributions and Functions of Random Variables$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.