The Poisson distribution models events occurring in fixed intervals, like defects in manufacturing or customer arrivals. It's characterized by lambda (λ), the average number of events per interval. This distribution is useful when events happen independently at a constant rate.

Calculating Poisson probabilities uses the probability mass function, which depends on lambda and the number of events. In business, it's applied to scenarios like predicting customer arrivals or defects in production. The Poisson distribution can also approximate the binomial distribution under certain conditions.

Poisson Distribution

Characteristics of Poisson distribution

• Discrete probability distribution models the number of events occurring in a fixed interval of time (hours) or space (area)
• Events occur independently of each other without influencing the occurrence of other events
• Rate at which events occur remains constant throughout the fixed interval
• Characterized by a single parameter, $\lambda$ (lambda) represents the average number of events per interval
  • Can be calculated as $\lambda = np$, where $n$ is the number of trials (50) and $p$ is the probability of success in each trial (0.02)

Probability calculation for Poisson events

• Poisson probability mass function (PMF): $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
  • $X$: Random variable representing the number of events (defective items)
  • $k$: Specific number of events (3 defective items)
  • $e$: Mathematical constant approximately 2.71828
  • $\lambda$: Average number of events per interval (2.5 defective items per batch)
  • $k!$: Factorial of $k$ $3! = 3 \times 2 \times 1 = 6$
• Substitute the values for $\lambda$ (2.5) and $k$ (3) into the PMF formula to calculate the probability

Poisson modeling in business scenarios

• Customer arrivals at a store or service center $\lambda$ represents the average number of customers arriving per hour (10)
  • Calculate the probability of a specific number of customers (15) arriving in a given hour
• Number of defects in a manufacturing process $\lambda$ represents the average number of defects per unit (0.1) or batch (5)
  • Calculate the probability of a specific number of defects (2) in a unit or batch
• Number of phone calls received by a call center $\lambda$ represents the average number of calls received per minute (4)
  • Calculate the probability of a specific number of calls (6) received in a given minute

Poisson vs binomial distributions

• Binomial distribution models the number of successes in a fixed number of independent trials (10)
  • Two possible outcomes: success (heads) or failure (tails)
  • Probability of success remains constant across trials (0.5)
• Poisson distribution can approximate the binomial distribution when:
  • The number of trials, $n$, is large typically greater than 20 (100)
  • The probability of success, $p$, is small typically less than 0.1 (0.02)
  • The product $np$ is less than 5 ($100 \times 0.02 = 2$)
• Under these conditions, the binomial distribution converges to the Poisson distribution with $\lambda = np$ ($\lambda = 100 \times 0.02 = 2$)