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Chapter 3: Problem 132

The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there are exactly five calls in one hour? (b) What is the probability that there are three or fewer calls in one hour? (c) What is the probability that there are exactly 15 calls in two hours? (d) What is the probability that there are exactly five calls in 30 minutes?

Short Answer

Expert verified
(a) 0.037, (b) 0.0102, (c) 0.0516, (d) 0.175.

Step by step solution

01

Understanding Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. For a Poisson random variable, \( X \), with an average rate of \( \lambda \), the probability of observing \( k \) events is given by the formula: \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \). In our problem, the average rate of calls per hour is \( \lambda = 10 \).
02

Calculate Probability for Exactly Five Calls in One Hour

We need to calculate \( P(X = 5) \) where \( X \) is the number of calls in one hour with \( \lambda = 10 \). Using the formula for the Poisson Distribution, \[ P(X = 5) = \frac{10^5 e^{-10}}{5!} = \frac{100000 e^{-10}}{120} \approx 0.037 \]
03

Calculate Probability for Three or Fewer Calls in One Hour

We need to find \( P(X \leq 3) \) for \( \lambda = 10 \).This can be calculated as:\[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]Calculating individually using the Poisson formula:- \( P(X = 0) = \frac{10^0 e^{-10}}{0!} = e^{-10} \approx 0.000045 \)- \( P(X = 1) = \frac{10^1 e^{-10}}{1!} = 10 e^{-10} \approx 0.00045 \)- \( P(X = 2) = \frac{10^2 e^{-10}}{2!} = 50 e^{-10} \approx 0.00225 \)- \( P(X = 3) = \frac{10^3 e^{-10}}{3!} = 166.67 e^{-10} \approx 0.0075 \)Adding them together,\[ P(X \leq 3) \approx 0.000045 + 0.00045 + 0.00225 + 0.0075 = 0.0102 \]
04

Calculate Probability for Exactly 15 Calls in Two Hours

For two hours, the average rate doubles \( \lambda = 20 \). We calculate \( P(X = 15) \) using the formula:\[ P(X = 15) = \frac{20^{15} e^{-20}}{15!} \approx 0.0516 \]
05

Calculate Probability for Exactly Five Calls in 30 Minutes

For 30 minutes, the average rate halves \( \lambda = 5 \). We then calculate \( P(X = 5) \) using the Poisson formula:\[ P(X = 5) = \frac{5^5 e^{-5}}{5!} = \frac{3125 e^{-5}}{120} \approx 0.175 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation involves determining the likelihood of an event occurring, using mathematical methods and formulas. When dealing with events over a fixed time or space, one effective way of calculating probability is by using statistical distributions like the Poisson distribution.
  • Poisson distribution helps us find the probability of a certain number of events happening in a defined interval.
  • The formula for Poisson probability is: \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \), where \( \lambda \) is the average number of occurrences.
In the context of the exercise, we use probability calculations to find out how likely it is for a specific number of telephone calls to occur within a given time frame. Constancy in \( \lambda \) showcases how predictable patterns of calls can aid in understanding the variability of these events.
Random Variables
In probability and statistics, random variables play an important role as they provide a numerical representation of outcomes from a random phenomenon.
  • A random variable assigns a numeric value to each possible outcome of a random event.
  • In our scenario, the number of telephone calls reaching a phone exchange can be expressed as a random variable \( X \).
These random variables can take discrete values, especially with distributions like the Poisson distribution, which specifically handles scenarios with events happening independently over a fixed time.
Event Modeling
Event modeling is used extensively in statistical analysis to represent the occurrence of events under certain conditions, like time intervals. By using models, we can predict and understand the behavior of complex events.
  • It allows the representation of actual events mathematically, facilitating probabilities and forecasts.
  • The Poisson model, in this case, helps to model the number of incoming calls based on the assumption that each call happens independently.
Using the model of random telephony events, we can calculate probabilities for different call volumes under varying conditions, like different time durations and different averages per hour, thus enabling effective predictive analytics and resource allocation.

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Most popular questions from this chapter

Determine the probability mass function for the random variable with the following cumulative distribution function: $$ F(x)=\left\\{\begin{array}{lr} 0 & x<2 \\ 0.2 & 2 \leq x<5.7 \\ 0.5 & 5.7 \leq x<6.5 \\ 0.8 & 6.5 \leq x<8.5 \\ 1 & 8.5 \leq x \end{array}\right. $$

A Web site randomly selects among 10 products to discount each day. The color printer of interest to you is discounted today. (a) What is the expected number of days until this product is again discounted? (b) What is the probability that this product is first discounted again exactly 10 days from now? (c) If the product is not discounted for the next five days, what is the probability that it is first discounted again 15 days from now? (d) What is the probability that this product is first discounted again within three or fewer days?

An automated egg carton loader has a \(1 \%\) probability of cracking an egg, and a customer will complain if more than one egg per dozen is cracked. Assume each egg load is an independent event. (a) What is the distribution of cracked eggs per dozen? Include parameter values. (b) What are the probability that a carton of a dozen eggs results in a complaint? c) What are the mean and standard deviation of the number of cracked eggs in a carton of one dozen?

A company employs 800 men under the age of 55 . Suppose that \(30 \%\) carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly one man has the marker? (b) If 10 men in the company are tested for the marker in this chromosome, what is the probability that more than one has the marker?

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