Calculating Poisson Distribution for Car Backfire Frequency on City Streets

In summary, the conversation involves using the poisson distribution to find the probability of hearing at most one car backfire in an hour, with a mean of 8 backfires per hour. The solution is found by adding the probability of hearing 0 backfires and 1 backfire, which is equal to e^(-8) + 8e^(-8) = 9e^(-8).
  • #1
buddingscientist
42
0
Hello In my text the following question is posed:

ON a city street, car backfires are heard 8 times per hour. Use the poisson distribution to find an exact expression for the prob. that a car backfire is heard at most once in a given hour. Do not simplify or evaluate your answer.


Now from my understanding, in the poisson distribution, mean = variance = 'parameter'

In the question we can assume that the mean equals 8, thus letting the parameter = 8.

A car backfire is heard at most once in a given hour.
= Probability of hearing 0 backfires + probability of hearing 1 backfire

= 8^0 e^(-8) / 0! + 8^1 e^(-8) / 1!
= e^(-8) + 8e^(-8)
= 9e^(-8)

Is this the correct solution? Thanks for your time
 
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  • #2
Yes!

-- AI
 
  • #3
Thanks very much
 

FAQ: Calculating Poisson Distribution for Car Backfire Frequency on City Streets

What is a Poisson distribution?

A Poisson distribution is a probability distribution that is used to model the number of times an event occurs in a given time interval or space, when the event is rare and random. It is named after French mathematician Siméon Denis Poisson.

What are the characteristics of a Poisson distribution?

A Poisson distribution is characterized by two parameters: the mean or average number of occurrences, and the time interval or space in which they occur. It is also a discrete distribution, meaning the number of occurrences can only be whole numbers.

What types of events can be modeled using a Poisson distribution?

Poisson distributions are commonly used to model events that are rare and random, such as the number of customers arriving at a store, the number of accidents on a highway, or the number of defects in a product. It can also be used to model the number of successes in a large number of trials, such as the number of heads when flipping a coin multiple times.

What is the formula for calculating the probability in a Poisson distribution?

The probability of a certain number of occurrences in a Poisson distribution can be calculated using the formula P(x) = (e^-λ * λ^x) / x!, where λ is the mean number of occurrences and x is the number of occurrences we are interested in.

How is a Poisson distribution different from a normal distribution?

While both Poisson and normal distributions are used to model data, they have different characteristics. Poisson distributions are discrete, while normal distributions are continuous. Poisson distributions are used for rare and random events, while normal distributions are used for more common events. Additionally, the shape of a Poisson distribution is skewed, while a normal distribution is symmetrical.

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