Poisson DistributionThe Poisson Distribution is used to model the number of events that occur in a time or space interval, when these events happen at a known average rate and are independent of the time since the last event. It is ideal for situations where rare events occur randomly.Poisson Distribution FormulaThe probability of exactly \( k \) events occurring in an interval is given by:\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]Where:- \( \lambda \): Average rate of occurrences in the interval.- \( k \): Desired number of events.- \( e \): Euler's number (approximately 2.71828).- \( k! \): Factorial of \( k \).Real-Life ExamplesExample 1: Phone calls- Situation: A call center receives, on average, 5 calls per hour.- Question: What is the probability of receiving exactly 3 calls in one hour?Solution:\[ \lambda = 5, \quad k = 3 \]\[ P(X = 3) = \frac{e^{-5} \cdot 5^3}{3!} = \frac{0.0067 \cdot 125}{6} \approx 0.1404 \quad (14.04\%) \]Example 2: Accidents on a highway- Situation: On a highway, an average of 2 accidents occur per day.- Question: What is the probability of exactly 4 accidents occurring in one day?Solution:\[ \lambda = 2, \quad k = 4 \]\[ P(X = 4) = \frac{e^{-2} \cdot 2^4}{4!} = \frac{0.1353 \cdot 16}{24} \approx 0.0902 \quad (9.02\%) \]Example 3: Defects on a production line- Situation: A factory produces, on average, 1 defective part every 10 hours.- Question: What is the probability of finding exactly 2 defective parts in 20 hours?Solution:\[ \lambda = 2 \quad (\text{20 hours} \div \text{10 hours per defect}) \]\[ P(X = 2) = \frac{e^{-2} \cdot 2^2}{2!} = \frac{0.1353 \cdot 4}{2} \approx 0.2707 \quad (27.07\%) \]Graph representing different types of PoissonPoisson ContextThe Poisson function (or Poisson distribution) emerged in the context of probability theory in the early 19th century, developed by the French mathematician Siméon Denis Poisson (1781–1840). It was introduced in his work "Research on the Probability of Judgments in Criminal and Civil Matters" (1837), where Poisson studied the probability of rare events in large populations, such as criminal convictions.The Poisson distribution emerged as an approximation of the binomial distribution when the number of trials (\( n \)) is large and the probability of success (\( p \)) is small, while keeping the product \( \lambda = n \cdot p \) constant. In other words, it models the occurrence of rare events in a fixed interval of time or space.When to Use Poisson and When to Use Binomial?The choice between using the Binomial Distribution and the Poisson Distribution depends on the characteristics of the problem you are analyzing. Both are used to model the number of events, but in different contexts.When to Use Binomial DistributionThe Binomial Distribution is used when:1. Fixed number of trials (\( n \)): The experiment consists of a fixed and known number of trials.2. Two possible outcomes: Each trial has only two possible outcomes: success or failure.3. Constant probability (\( p \)): The probability of success (\( p \)) is the same in each trial.4. Independent trials: The result of one trial does not affect the result of the others.Application examples:- Tossing a coin \( n \) times and counting the number of heads.- Testing \( n \) products and counting how many are defective.- Performing \( n \) attempts to shoot at goal and counting how many goals are scored.When to Use Poisson DistributionThe Poisson Distribution is used when:1. Rare events: The event of interest is rare in relation to the time or space interval.2. Known average rate (\( \lambda \)): The average number of occurrences (\( \lambda \)) in an interval is known.3. Independent events: The occurrence of one event does not affect the occurrence of others.4. Continuous interval: Events can occur at any point in a continuous interval (time, space, etc.).Application examples:- Number of calls received at a call center per hour.- Number of accidents on a highway per day.- Number of typos on a page of text.- Number of customers arriving at a restaurant per minute.Exercises1. Phone calls:- A call center receives, on average, 4 calls per hour. What is the probability of receiving exactly 6 calls in one hour?2. Accidents on a highway:- On a highway, an average of 3 accidents occur per day. What is the probability of exactly 5 accidents occurring in one day?3. Defects on a production line:- A factory produces, on average, 2 defective parts every 8 hours. What is the probability of finding exactly 3 defective parts in 24 hours?4. Customer arrivals:- A restaurant receives, on average, 10 customers per hour. What is the probability of receiving exactly 12 customers in one hour?5. Emails received:- An employee receives, on average, 8 emails per hour. What is the probability of receiving exactly 10 emails in one hour?6. System failures:- A system fails, on average, once per day. What is the probability that the system fails exactly 2 times in one day?7. Births in a hospital:- A hospital records, on average, 5 births per day. What is the probability of exactly 7 births occurring in one day?8. Cars passing through a tollbooth:- A tollbooth records, on average, 20 cars per minute. What is the probability of exactly 25 cars passing through in one minute?9. Typos:- A typist makes, on average, 3 errors per page. What is the probability of making exactly 5 errors on one page?10. Rain in a city:- In a city, it rains, on average, 4 days per month. What is the probability of raining exactly 6 days in one month?Answer Key1. 10.42%2. 10.08%3. 8.92%4. 9.48%5. 9.93%6. 18.39%7. 10.44%8. 4.46%9. 10.08%10. 10.42%
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