3.1 Discrete probability distributions

Discrete probability distributions are powerful tools for modeling real-world events with finite outcomes. They help us understand and predict everything from coin flips to customer arrivals, assigning probabilities to each possible result.

Key distributions like Bernoulli, binomial, Poisson, and geometric have unique properties and applications. By calculating probabilities, expected values, and variances, we can make informed decisions in areas like quality control, marketing, and finance.

Fundamentals of Discrete Probability Distributions

Define and explain discrete probability distributions

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• Discrete probability distribution assigns probabilities to discrete random variables representing likelihood of outcomes in finite or countable set
• Key characteristics encompass non-negative probabilities, sum of all probabilities equaling 1, and mutually exclusive outcomes
• Common examples include binomial distribution (coin flips), Poisson distribution (customer arrivals), and geometric distribution (waiting time)

Identify and describe the properties of common discrete probability distributions

• Bernoulli distribution models single trial with two possible outcomes (success or failure) with parameter p (probability of success), mean $\mu = p$, and variance $\sigma^2 = p(1-p)$
• Binomial distribution models number of successes in n independent Bernoulli trials with parameters n (number of trials) and p (probability of success), mean $\mu = np$, and variance $\sigma^2 = np(1-p)$
• Poisson distribution models number of events occurring in fixed interval with parameter λ (average rate of occurrence), mean and variance $\mu = \sigma^2 = \lambda$
• Geometric distribution models number of trials until first success with parameter p (probability of success), mean $\mu = \frac{1}{p}$, and variance $\sigma^2 = \frac{1-p}{p^2}$

Applications and Calculations

Calculate probabilities using discrete probability distributions

• Probability mass function gives probability of each possible outcome
  • Binomial PMF: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
  • Poisson PMF: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$
• Cumulative distribution function gives probability of random variable being less than or equal to value by summing PMF values up to desired point
• Expected value calculated as $E(X) = \sum_{x} x \cdot P(X=x)$
• Variance calculated as $Var(X) = E(X^2) - [E(X)]^2$

Apply discrete probability distributions to real-world scenarios

• Binomial distribution applications include quality control (defective items in production batch), marketing (success rate of email campaigns), and finance (successful trades in day)
• Poisson distribution applications encompass customer service (calls received per hour), inventory management (items sold per day), and insurance (claims filed in month)
• Geometric distribution applications involve manufacturing (items inspected before finding defect), sales (calls made before securing deal), and research (trials before successful experiment)

Interpret the results of probability calculations in a business context

• Decision-making based on probabilities involves risk assessment in project management, resource allocation in operations, and forecasting in supply chain management
• Performance evaluation compares actual outcomes to expected probabilities and identifies areas for improvement or investigation
• Scenario analysis uses probability distributions to model different business outcomes and assess likelihood of various scenarios for strategic planning