Statistics - Poisson distribution question.

In summary, the question is asking for the probability of having to wait at least one more minute before the first call is received given that no call was received during the first 3 minutes. The solution is P(T≥1) = 1 - FT(1) = e-1/3. This is because you want to find the probability of the event {T > 4} given that the event {T > 3} has already occurred.
  • #1
peripatein
880
0
Hi,

Homework Statement


I am somewhat perplexed by the proposed solution to the following Statistics problem and was hoping one of you might be willing to help me settle this:
An operator receives phone calls between 8AM and 4PM at an average rate of 20 calls/hour. No call was received during the first 3 minutes. What is the probability that we shall have to wait at least one more minute before the first call is received?

Homework Equations


The Attempt at a Solution


Now, the book states it ought to be P(T≥1) = 1 - FT(1) = e-1/3.
My question is, why should it not have been 1 - [F(4) - F(3)], i.e. the complement of a call received between the 3rd and 4th minutes?
I'd appreciate it if any of you could explain this to me.
 
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  • #2
peripatein said:
Hi,

Homework Statement


I am somewhat perplexed by the proposed solution to the following Statistics problem and was hoping one of you might be willing to help me settle this:
An operator receives phone calls between 8AM and 4PM at an average rate of 20 calls/hour. No call was received during the first 3 minutes. What is the probability that we shall have to wait at least one more minute before the first call is received?


Homework Equations





The Attempt at a Solution


Now, the book states it ought to be P(T≥1) = 1 - FT(1) = e-1/3.
My question is, why should it not have been 1 - [F(4) - F(3)], i.e. the complement of a call received between the 3rd and 4th minutes?
I'd appreciate it if any of you could explain this to me.

If T is the time of the first arrival, you want P{T > 4 | T > 3}, because you are told that the event {T > 3} occurred.
 
  • #3
Thank you, Ray! It's clearer now :-).
 

1. What is a Poisson distribution?

A Poisson distribution is a probability distribution that is used to model the number of events that occur within a specific time interval or space, when the events are independent of each other and the average rate of occurrence is known. It is characterized by a single parameter, λ (lambda), which represents the average rate of occurrence.

2. How is the Poisson distribution different from other probability distributions?

The Poisson distribution differs from other distributions, such as the normal distribution, in that it is used for discrete random variables (i.e. count data) rather than continuous data. It also assumes that the events occur independently of each other and at a constant rate, whereas other distributions may have different assumptions.

3. What types of data are suitable for modeling with a Poisson distribution?

The Poisson distribution is commonly used to model count data, such as the number of customers that enter a store, the number of accidents that occur in a day, or the number of goals scored in a soccer game. It is also used to model rare events, where the probability of occurrence is low but the number of opportunities for occurrence is large.

4. How is the Poisson distribution calculated?

The probability of a specific number of events occurring in a given time interval or space can be calculated using the Poisson probability mass function. The formula is P(X=x) = (e^-λ * λ^x) / x!, where x is the number of events, λ is the average rate of occurrence, and e is the mathematical constant approximately equal to 2.71828.

5. What are some real-life applications of the Poisson distribution?

The Poisson distribution has many real-life applications, including in finance, biology, and sports. For example, it can be used to model stock market fluctuations, the number of mutations in a DNA sequence, and the number of goals scored by a soccer team in a season. It is also commonly used in the insurance industry to model the number of claims filed by policyholders.

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