Poisson distribution and binomial distribution questions

In summary, the conversation discusses the Poisson and binomial probability distributions, as well as a specific example involving defective nails and lawn mower rentals. The conversation also mentions the rules of the forum and the limitations on providing help with homework problems.
  • #1
latkan
5
0
Please help with this thanks :)

1.
(a) Define the Poisson probability distribution with mean μ.

(b) Write down the binomial distribution for x successes in n independent trials each with probability p of success.

(c) On average, 0.15% of the nails manufactured at a factory are known to be defective. If a random sample of 400 nails is inspected, what is the probability of there being no more than 3 defective nails?

4.
(a) Define the Poisson probability distribution with mean p.

(b) A tool hire shop has six lawn mowers which it hires out on a daily basis.The number of lawn mowers requested per day follows a Poisson probability distribution with mean 4.5. Find the probability that:

(i) exactly three lawn mowers are hired out on anyone day;
(ii) all lawn mowers are in use on anyone day.
 
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  • #2
1. This isn't number theory.
2. This forum isn't for homework; homework forums are at the top.
3. We can't help you with homework problems unless you show your own work.
 

1. What is the difference between Poisson distribution and binomial distribution?

Poisson distribution is used to model the probability of a certain number of events occurring within a specific period of time, assuming a constant rate of occurrence. Binomial distribution, on the other hand, is used to model the probability of a certain number of successes in a fixed number of trials, where each trial has a constant probability of success.

2. How is the mean calculated for Poisson and binomial distributions?

For Poisson distribution, the mean is equal to the rate parameter λ, which represents the average number of events occurring in a given time period. For binomial distribution, the mean is equal to the product of the number of trials (n) and the probability of success (p).

3. Can Poisson or binomial distributions be used to model continuous data?

No, both distributions are used for discrete data, where the outcome can only take on a finite or countable number of values. For continuous data, other distributions such as normal or exponential may be more appropriate.

4. What is the relationship between Poisson and binomial distributions?

A binomial distribution can be approximated by a Poisson distribution when the number of trials is large and the probability of success is small. This is known as the Poisson approximation to the binomial distribution.

5. How is the Poisson distribution used in real-world applications?

The Poisson distribution is commonly used in fields such as epidemiology, finance, and telecommunications to model the occurrence of rare events, such as disease outbreaks, stock market crashes, or dropped calls in a phone network. It can also be used to estimate the number of customers arriving at a store or the number of defects in a manufacturing process.

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