Poisson Distribution: The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that the events occur at a constant average rate and are independent of each other. It is often used when dealing with rare events or when the probability of an event occurring is low. The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of event occurrence.

Properties and applications of the Poisson distribution:

1. Probability Mass Function: The probability mass function (PMF) of the Poisson distribution is given by P(X = k) = (e^(-λ) * λ^k) / k!, where X represents the random variable, k represents the number of events, and λ is the average rate of event occurrence.
2. Mean and Variance: The mean of the Poisson distribution is equal to λ, and the variance is also equal to λ. This means that the distribution is fully defined by its average rate parameter.
3. Applications: The Poisson distribution is commonly used in various fields, including:
   • Modeling the number of customer arrivals in a given time period.
   • Analyzing the number of phone calls received per hour.
   • Predicting the number of accidents in a specific location over a period of time.
   • Studying the number of defective items in a manufacturing process.
   • Analyzing the number of emails received per day.

Normal Distribution (Gaussian Distribution): The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is widely used in statistics. It is characterized by its bell-shaped curve and is symmetrical around the mean. Many natural phenomena and statistical measurements follow a normal distribution.

Properties and applications of the normal distribution:

1. Probability Density Function: The probability density function (PDF) of the normal distribution is given by the formula: f(x) = (1 / sqrt(2πσ^2)) * e^(-(x – μ)^2 / (2σ^2)), where x represents the random variable, μ is the mean, and σ is the standard deviation.
2. Mean and Variance: The mean of a normal distribution is equal to μ, and the variance is equal to σ^2. The standard deviation (σ) determines the spread or dispersion of the distribution.
3. Empirical Rule: The normal distribution follows the empirical rule, also known as the 68-95-99.7 rule, which states that:
   • Approximately 68% of the data falls within one standard deviation of the mean.
   • Approximately 95% of the data falls within two standard deviations of the mean.
   • Approximately 99.7% of the data falls within three standard deviations of the mean.
4. Applications: The normal distribution is widely used in various fields, including:
   • Modeling continuous variables such as height, weight, and IQ scores.
   • Hypothesis testing and confidence interval estimation.
   • Statistical process control.
   • Financial analysis and risk management.
   • Quality control and manufacturing processes.

The Poisson distribution is suitable for modeling discrete events that occur randomly in time or space, while the normal distribution is used to model continuous variables that follow a symmetric bell-shaped pattern. Both distributions have distinct characteristics and applications, and they play a crucial role in statistical analysis and probability theory.