Discrete distributions are the building blocks of probability theory. They help us model real-world events with specific outcomes. From coin flips to disease outbreaks, these distributions give us tools to understand and predict random phenomena.

This section covers four key discrete distributions: Bernoulli, binomial, geometric, and Poisson. We'll explore their properties, calculations, and real-world applications. Understanding these distributions is crucial for tackling probability problems in various fields.

Key Properties of Discrete Distributions

Bernoulli and Binomial Distributions

• The Bernoulli distribution is a discrete probability distribution for a random variable that takes on only two possible values, typically denoted as 1 (success) and 0 (failure), with a probability of success $p$
• The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success $p$
• The probability mass function (PMF) for the Bernoulli distribution is $P(X = x) = p^x (1-p)^(1-x)$, where $x \in {0, 1}$
• The PMF for the binomial distribution is $P(X = k) = \binom{n}{k} * p^k * (1-p)^(n-k)$, where $k \in {0, 1, ..., n}$

Geometric and Poisson Distributions

• The geometric distribution is a discrete probability distribution that models the number of independent Bernoulli trials needed to achieve the first success, each with the same probability of success $p$
• The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given an average rate $\lambda$ and the assumption that events occur independently
• The PMF for the geometric distribution is $P(X = k) = (1-p)^(k-1) p$, where $k \in {1, 2, ...}$
• The PMF for the Poisson distribution is $P(X = k) = (\lambda^k e^{-\lambda}) / k!$, where $k \in {0, 1, 2, ...}$

Probability Calculations for Discrete Distributions

Expected Values and Variances

• The expected value (mean) for the Bernoulli distribution is $E(X) = p$
• The expected value for the binomial distribution is $E(X) = n p$
• The expected value for the geometric distribution is $E(X) = 1 / p$
• The expected value for the Poisson distribution is $E(X) = \lambda$
• The variance for the Bernoulli distribution is $Var(X) = p (1-p)$
• The variance for the binomial distribution is $Var(X) = n * p * (1-p)$
• The variance for the geometric distribution is $Var(X) = (1-p) / p^2$
• The variance for the Poisson distribution is $Var(X) = \lambda$

Calculating Probabilities for Specific Events

• To calculate probabilities for specific events, use the PMF of the appropriate distribution and sum the probabilities of the desired outcomes
• For the binomial distribution, the cumulative distribution function (CDF) can be used to calculate probabilities of the form $P(X \leq k)$ or $P(X > k)$
• Example: In a binomial distribution with $n=10$ and $p=0.3$, calculate $P(X \leq 2)$ using the CDF
• Example: In a Poisson distribution with $\lambda=3$, calculate $P(X=4)$ using the PMF

Applications of Discrete Distributions

Quality Control and Finance

• In quality control, the binomial distribution can be used to model the number of defective items in a batch of products
  • Example: A manufacturer produces batches of 100 items, and each item has a 2% chance of being defective. The binomial distribution can be used to calculate the probability of having a specific number of defective items in a batch
• In finance, the geometric distribution can be used to model the number of trading days until a stock price reaches a certain level
  • Example: If a stock has a 45% chance of increasing each day, the geometric distribution can be used to calculate the probability of the stock reaching a specific price level within a given number of trading days

Biology, Telecommunications, and Epidemiology

• In biology, the Poisson distribution can be used to model the number of mutations in a DNA sequence over a fixed length
  • Example: If mutations occur at an average rate of 0.001 per base pair, the Poisson distribution can be used to calculate the probability of observing a specific number of mutations in a 1000 base pair sequence
• In telecommunications, the Poisson distribution can be used to model the number of phone calls arriving at a call center within a given time frame
  • Example: If a call center receives an average of 20 calls per hour, the Poisson distribution can be used to calculate the probability of receiving a specific number of calls within a 30-minute period
• In epidemiology, the Poisson distribution can be used to model the number of disease cases in a population over a specific period
  • Example: If a rare disease affects an average of 5 people per 100,000 population per year, the Poisson distribution can be used to calculate the probability of observing a specific number of cases in a city of 500,000 people over a 6-month period

Discrete Distributions vs Real-World Phenomena

Bernoulli and Binomial Distributions

• The Bernoulli distribution is appropriate when an experiment has only two possible outcomes (success or failure) and the probability of success remains constant for each trial
  • Example: Flipping a fair coin (heads or tails) or determining whether a manufactured item is defective (defective or non-defective)
• The binomial distribution is appropriate when an experiment consists of a fixed number of independent Bernoulli trials, each with the same probability of success
  • Example: The number of successful free throws out of 10 attempts, given a player's free throw success probability

Geometric and Poisson Distributions

• The geometric distribution is appropriate when an experiment consists of independent Bernoulli trials, each with the same probability of success, and the goal is to determine the number of trials needed to achieve the first success
  • Example: The number of job interviews a candidate attends before receiving a job offer, given a fixed probability of success for each interview
• The Poisson distribution is appropriate when modeling the number of rare events occurring in a fixed interval of time or space, assuming events occur independently and at a constant average rate
  • Example: The number of earthquakes in a region over a year, given the average rate of occurrence
  • Example: The number of typing errors per page in a book, given the average error rate per page