Poisson, Binomial Distributions

In summary, a Poisson distribution is a probability distribution used to model the number of events in a fixed time or space, while a Binomial distribution models the number of successes in a fixed number of trials. Both distributions have different assumptions and are used in various real-life applications. The mean and standard deviation for a Poisson distribution are equal to the average rate of occurrence and its square root, respectively, while for a Binomial distribution, the mean is equal to the product of the number of trials and the probability of success, and the standard deviation is equal to the square root of this product.
  • #1
Millacol88
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Homework Statement



The number of claims that an insurance company receives per week is a random variable with the Poisson distribution with parameter λ. The probability that a claim will be accepted as genuine is p, and is independent of other claims.

a) What is the probability that no claim will be accepted over one week?
b) Find the expected number of accepted claims over one week.
c) Let N be the number of accepted claims over one week. Find the probability distribution for N.

Homework Equations



Poisson: P(X=x) = λx/x! e

The Attempt at a Solution



a) If x is the number of attempted claims, P(N = 0) = (1 - p)x, I think.

b/c) The distribution for N should be binomial(x, p). Now the expectation value of X is λ. And the expectation value of N is xp. Would this then become λp?
 
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  • #2
Millacol88 said:

Homework Statement



The number of claims that an insurance company receives per week is a random variable with the Poisson distribution with parameter λ. The probability that a claim will be accepted as genuine is p, and is independent of other claims.

a) What is the probability that no claim will be accepted over one week?
b) Find the expected number of accepted claims over one week.
c) Let N be the number of accepted claims over one week. Find the probability distribution for N.

Homework Equations



Poisson: P(X=x) = λx/x! e

The Attempt at a Solution



a) If x is the number of attempted claims, P(N = 0) = (1 - p)x, I think.

b/c) The distribution for N should be binomial(x, p). Now the expectation value of X is λ. And the expectation value of N is xp. Would this then become λp?

Yes.

I mean: the expectation is ##\lambda p##. You still need to deal with the issue of the probability values---not just the expected value.
 
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FAQ: Poisson, Binomial Distributions

What is a Poisson distribution?

A Poisson distribution is a probability distribution that is used to model the number of events that occur in a fixed time interval or space, given the average rate of occurrence and the assumption that each event is independent of the others.

What is a Binomial distribution?

A Binomial distribution is a probability distribution that is used to model the number of successes in a fixed number of independent trials, given a constant probability of success for each trial.

What is the difference between a Poisson and a Binomial distribution?

The main difference between a Poisson and a Binomial distribution is that a Poisson distribution models the number of events in a fixed time or space, while a Binomial distribution models the number of successes in a fixed number of trials. Additionally, a Poisson distribution assumes that the events occur randomly and independently, while a Binomial distribution assumes a fixed probability of success for each trial.

What are some real-life applications of Poisson and Binomial distributions?

Poisson distributions are often used in fields such as biology, epidemiology, and telecommunications to model the number of events or occurrences in a given time or space. Binomial distributions are commonly used in areas such as quality control, market research, and sports analytics to model the number of successes in a fixed number of trials.

How do you calculate the mean and standard deviation of a Poisson or Binomial distribution?

The mean of a Poisson distribution is equal to the average rate of occurrence, and the standard deviation is equal to the square root of the mean. For a Binomial distribution, the mean is equal to the number of trials multiplied by the probability of success, and the standard deviation is equal to the square root of the product of the number of trials and the probability of success.

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