📊Probabilistic Decision-Making Unit 3 – Discrete & Continuous Distributions

Key Concepts and Definitions

• Probability distributions describe the likelihood of different outcomes in a random experiment or process
• Random variables represent the possible outcomes of a random event and can be discrete (countable) or continuous (uncountable)
• Probability mass functions (PMFs) define the probability of each possible value for a discrete random variable
• Probability density functions (PDFs) specify the relative likelihood of a continuous random variable taking on a specific value
• Cumulative distribution functions (CDFs) give the probability that a random variable is less than or equal to a particular value
• Expected value is the average outcome of a random variable over many trials, calculated as the sum or integral of each value multiplied by its probability
• Variance and standard deviation measure the spread or dispersion of a probability distribution around its expected value

Types of Probability Distributions

• Discrete distributions have a countable number of possible outcomes, such as the number of defective items in a batch (Bernoulli, binomial, Poisson)
• Continuous distributions have an uncountable, infinite number of possible outcomes within a range, like the time until a machine fails (normal, exponential, uniform)
• Joint distributions describe the probabilities of two or more random variables occurring together
• Marginal distributions are derived from joint distributions by summing or integrating over the other variables
• Conditional distributions give the probabilities of one variable given specific values of another
• Multivariate distributions involve multiple random variables, which can be discrete, continuous, or a mix of both
• Some distributions can be either discrete or continuous depending on the context and assumptions (geometric, negative binomial)

Discrete Distributions

• Bernoulli distribution models a single trial with two possible outcomes (success or failure), with probability of success $p$
• Binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, with parameters $n$ (trials) and $p$ (success probability)
  • PMF: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ for $k = 0, 1, ..., n$
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space, with rate parameter $\lambda$
  • PMF: $P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$ for $k = 0, 1, 2, ...$
• Geometric distribution gives the number of trials until the first success in a series of independent Bernoulli trials
• Negative binomial distribution extends the geometric to the number of trials until the $r$-th success
• Hypergeometric distribution describes the number of successes in a fixed number of draws from a finite population without replacement

Continuous Distributions

• Normal (Gaussian) distribution is bell-shaped and symmetric, defined by mean $\mu$ and standard deviation $\sigma$
  • PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$ for $-\infty < x < \infty$
  • Standard normal has $\mu=0$ and $\sigma=1$, with CDF denoted $\Phi(z)$
• Exponential distribution models the time between events in a Poisson process, with rate parameter $\lambda$
  • PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
  • Memoryless property: $P(X>s+t|X>s) = P(X>t)$ for all $s,t \geq 0$
• Uniform distribution has equal probability over a continuous range $[a,b]$
  • PDF: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
• Gamma distribution generalizes the exponential, with shape $\alpha$ and rate $\beta$
• Beta distribution is defined on $[0,1]$ with shape parameters $\alpha$ and $\beta$, used for prior and posterior distributions in Bayesian inference
• Chi-square, Student's t, and F distributions are related to the normal and used in hypothesis testing and confidence intervals

Properties and Parameters

• Parameters define the shape, center, and spread of a distribution
  • Location parameters shift the distribution (mean, median, mode)
  • Scale parameters stretch or compress the distribution (variance, standard deviation)
  • Shape parameters affect the overall form (skewness, kurtosis)
• Moment generating functions (MGFs) uniquely characterize a distribution and can derive moments like expected value and variance
• Transformations of random variables lead to new distributions
  • Linear transformations $Y=aX+b$ change location and scale
  • Functions of random variables have distributions derived using change of variables techniques
• Relationships between distributions enable analysis and problem-solving
  • Central Limit Theorem: Sums of independent random variables converge to a normal distribution
  • Poisson approximation to binomial for rare events
  • Exponential as a special case of gamma; chi-square as a sum of squared standard normals
• Inequalities and bounds constrain probabilities for any distribution
  • Markov's inequality bounds the probability of a non-negative random variable exceeding a value
  • Chebyshev's inequality limits the probability outside a certain number of standard deviations from the mean

Probability Calculations

• For discrete distributions, probabilities are summed over the values of interest
  • $P(a \leq X \leq b) = \sum_{x=a}^b P(X=x)$
• For continuous distributions, probabilities are integrals of the PDF over the region of interest
  • $P(a \leq X \leq b) = \int_a^b f(x) dx$
• Complements: $P(X>a) = 1 - P(X\leq a)$
• Union of disjoint events: $P(X=a \text{ or } X=b) = P(X=a) + P(X=b)$ for $a \neq b$
• Conditional probability: $P(X=a|Y=b) = \frac{P(X=a,Y=b)}{P(Y=b)}$ if $P(Y=b)>0$
• Independence: $X$ and $Y$ are independent if $P(X=a,Y=b)=P(X=a)P(Y=b)$ for all $a,b$
  • For independent $X$ and $Y$, $E(XY)=E(X)E(Y)$ and $\text{Var}(X+Y) = \text{Var}(X)+\text{Var}(Y)$
• Total Probability Theorem: $P(B) = \sum_i P(A_i)P(B|A_i)$ for a partition ${A_i}$
• Bayes' Theorem: $P(A_i|B) = \frac{P(A_i)P(B|A_i)}{\sum_j P(A_j)P(B|A_j)}$ for a partition ${A_i}$

Applications in Decision-Making

• Distributions model uncertainty and variability in real-world systems
  • Demand forecasting: Poisson for low-volume products, normal for high-volume
  • Reliability engineering: Exponential for constant failure rates, Weibull for time-varying
  • Quality control: Binomial for pass/fail inspection, normal for continuous measurements
• Expected values guide decisions based on long-run averages
  • Expected profit maximization
  • Expected utility theory for preferences under risk
• Quantiles and percentiles inform risk management
  • Value-at-Risk (VaR) for financial portfolios
  • Safety stock levels for inventory management
• Bayesian inference updates prior beliefs with data to make better decisions
  • Conjugate priors (beta-binomial, gamma-Poisson) simplify calculations
  • Posterior distributions combine prior knowledge with observed evidence
• Simulation and resampling methods (Monte Carlo, bootstrap) enable complex modeling and risk assessment
• Stochastic optimization techniques (newsvendor, multi-armed bandits) balance exploration and exploitation in the face of uncertainty

Common Pitfalls and Misconceptions

• Assuming normality without justification; many real-world phenomena have skewed or heavy-tailed distributions
• Misinterpreting probability density as probability; PDFs can exceed 1 but probabilities cannot
• Confusing independence and mutual exclusivity; disjoint events can be dependent
• Neglecting to normalize probabilities to sum or integrate to 1
• Misapplying memoryless property; only holds for exponential and geometric distributions
• Incorrectly extending properties of one distribution to others without proof
• Ignoring or mishandling outliers and extreme values; they can significantly impact decisions and risk assessments
• Overreliance on point estimates; consider full distributions and intervals for uncertainty quantification