Probability of 1 Phone Call: Solve w/ Exponential Distrib.

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In summary, the conversation discusses the probability of receiving exactly one phone call in one night, given that there is an average of one call per night. The probability is shown to be e^-1 under the assumption of a Poisson distribution, which can be derived from the binomial distribution. This is a topic covered in Mathematical Statistics 2.
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Providence88
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Homework Statement



Suppose that on average, I will receive one phone call every night. Under reasonable assumptions (not given in this question prompt), show that the probability of receiving exactly one phone call, tonight, is e^-1.

Homework Equations



Possibly the gamma distribution function or exponential distribution function, but I'm not sure, as this problem arises out of material far out of the introductory distribution functions.

That is,

f(y) = [tex]\frac{1}{\Gamma(\alpha)\beta^{\alpha}} * y^{\alpha-1}e^{\frac{-y}{\beta}[/tex]

is a gamma distribution for y>0. An exponential distribution, of course, is just a gamma with alpha = 1.

The Attempt at a Solution



Well, I thought that if [tex]\beta = 1[/tex], then you simply compute f(1), where y is the number of calls in a day. This, however, seems too easy, considering that this class is Mathematical Statistics 2, and we learned that sort of thing in M.S. 1.

Thanks!
 
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  • #2
I think you are supposed to derive the probability without using a given arbitrary distribution. I would guess that the reasonable assumption is that there is a pool of N people that might call you, each one with probability p. So just use the binomial distribution. Now let N go to infinity.
 
  • #3
Dick said:
I think you are supposed to derive the probability without using a given arbitrary distribution. I would guess that the reasonable assumption is that there is a pool of N people that might call you, each one with probability p. So just use the binomial distribution. Now let N go to infinity.
I guess you're talking about Poisson approximation? I had to look that one up. It's mentioned in the book, but not under that term. Very clever. I'll have to try that out.
 
  • #4
Yes, I am. But you shouldn't need to look it up. I think they want you to derive it from the binomial distribution. That would be worthy of a Stat 2 exercise.
 
  • #5
Right. I'll try and derive it without looking back at the book. Thanks!
 

What is the probability of receiving exactly 1 phone call in a given time period?

The probability of receiving exactly 1 phone call in a given time period can be calculated using the exponential distribution formula. This formula takes into account the average rate of phone calls and the length of the time period. The resulting value will be the probability of receiving 1 phone call during that time period.

How is the exponential distribution used to model the probability of 1 phone call?

The exponential distribution is used to model the probability of 1 phone call by assuming that phone calls occur randomly and independently over time. This means that the probability of a phone call occurring at any given moment is constant and unaffected by previous calls. The exponential distribution formula takes into account this assumption to calculate the probability of 1 phone call.

What is the relationship between the average rate of phone calls and the probability of 1 phone call?

The average rate of phone calls is directly related to the probability of 1 phone call. As the average rate of phone calls increases, the probability of receiving 1 phone call also increases. This is because a higher average rate means more phone calls are expected to occur during a given time period, resulting in a higher probability of receiving 1 phone call.

How can the exponential distribution be used to predict the likelihood of multiple phone calls?

The exponential distribution can be used to predict the likelihood of multiple phone calls by adjusting the formula to calculate the probability of receiving 2 or more calls. This can be useful for businesses or call centers to estimate the number of staff needed to handle a certain number of phone calls during a given time period.

What are the limitations of using the exponential distribution to model the probability of 1 phone call?

One limitation of using the exponential distribution to model the probability of 1 phone call is that it assumes phone calls occur randomly and independently. In reality, phone calls may be affected by external factors such as holidays, events, or marketing campaigns. Additionally, the distribution assumes a constant average rate of phone calls, which may not always be the case. Other factors such as time of day or day of the week may also impact the probability of receiving 1 phone call. Therefore, the exponential distribution should be used with caution and other factors should be taken into consideration when predicting the probability of 1 phone call.

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