Study Notes on Binomial and Poisson Distributions

Overview of Probability Distributions

Definition: The Binomial Distribution describes the number of successes in a fixed number of trials in a binomial experiment, where each trial has two possible outcomes (success or failure).
Probability Mass Function (PMF): Denotes the probability of observing exactly $k$ successes in $n$ trials.
- The PMF is given by the formula:
  P(X = k) = {n rack k} p^k (1-p)^{n-k}
  where:
- ${n rack k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials.
- $p$ is the probability of success on each trial.
- $1 - p$ is the probability of failure on each trial.
Cumulative Distribution Function (CDF): Summation of probabilities from $0$ to $k$, representing at most $k$ successes.
Symmetry: The PMF shows symmetry around its center, illustrated in plots.

Practical Application: Example of defective items with a defect probability of $0.01$. Assumption is that products are independent.
- Risk Management: Companies need to evaluate the probability of defect rates to avoid excessive returns which could harm reputation and trust.
- If 10 items are tested, calculate the probability of at least one defect using the complement: 1 minus the probability of having zero defects.
Complements in Calculations:
- To find the probability of having at least one defective item, use:
  P(X \geq 1) = 1 - P(X = 0)
- Calculating $P(X = 0)$ gives rise to the formula:
  P(X = 0) = (1 - p)^{n}
Example Result: The probability of having to replace at least one defective item out of 10 items tested is calculated to be approximately $0.004$ or $0.4$%.

Expectation (Mean): In a binomial distribution, the expected number of successes is given by:
E(X) = n \cdot p
Variance: The variance is given by:
Var(X) = n \cdot p \cdot (1-p)
In practice, managers use these calculations to budget resources for customer service and manage risk effectively.

Using the Expectation Formula:
- The expectation value requires summing across all possible values of $k$ from $0$ to $n$ multiplied by their respective probabilities:
  E(X) = \sum_{k=0}^{n} k \cdot P(X = k)
The challenge lies in computing this summation which may take time and effort.
Simplified Case: For a single trial ($n=1$), then:
- E(X) = p
- Var(X) = p(1-p) as an illustrative example for manual calculations.

Poisson Distribution: Describes the probability of a given number of events occurring in a fixed interval of time or space; does not limit the maximum successes to a finite number like the binomial.
- The PMF of the Poisson Distribution is given by:
  P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} where $eta$ is the expected number of occurrences in the interval.

Applications in Statistics:
- To make inferences based on empirical data regarding probabilities—mean and variance facilitate statistical conclusions and facilitate effective planning by management.
Tools and Software:
- R, Python, and other software are suggested for performing calculations and drawing distributions, emphasizing their importance in current statistical practices.
Learning Resources: Encouragement for future research and cross-disciplinary collaboration using statistical models to contribute effectively in real-world scenarios.