13.1 Discrete probability distributions in combinatorics

Discrete probability distributions are the building blocks of random variable analysis in combinatorics. They help us model and understand events with distinct outcomes, like coin flips or dice rolls, by assigning probabilities to each possible result.

This section covers key distributions like Bernoulli, Binomial, Poisson, and Geometric. We'll explore their properties, applications, and how to calculate important values like expected outcomes and variability. Understanding these concepts is crucial for tackling real-world probability problems.

Probability Distributions

Bernoulli and Binomial Distributions

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• Bernoulli distribution models single binary outcome experiments (success or failure)
• Probability mass function for Bernoulli distribution given by $P(X=k) = p^k(1-p)^{1-k}$ where k is 0 or 1
• Binomial distribution extends Bernoulli to n independent trials
• Probability mass function for Binomial distribution expressed as $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$
• Applications include coin flips (heads or tails) and quality control (defective or non-defective items)
• Parameters for Binomial distribution include n (number of trials) and p (probability of success)
• Mean of Binomial distribution calculated as $[E(X)](https://www.fiveableKeyTerm:e(x)) = np$
• Variance of Binomial distribution determined by $[Var(X)](https://www.fiveableKeyTerm:var(x)) = np(1-p)$

Poisson and Geometric Distributions

• Poisson distribution models rare events in a fixed interval
• Probability mass function for Poisson distribution defined as $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
• λ (lambda) represents average number of events in the interval
• Applications encompass radioactive decay and customer arrivals at a store
• Geometric distribution models number of trials until first success
• Probability mass function for Geometric distribution given by $P(X=k) = (1-p)^{k-1}p$
• p denotes probability of success on each trial
• Mean of Geometric distribution calculated as $E(X) = \frac{1}{p}$
• Variance of Geometric distribution determined by $Var(X) = \frac{1-p}{p^2}$
• Applications include number of coin flips until first heads or attempts to win a game

Hypergeometric Distribution

• Hypergeometric distribution models sampling without replacement from finite population
• Probability mass function expressed as $P(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$
• N represents total population size, K denotes number of success states in population
• n signifies sample size, k indicates number of successes in sample
• Differs from Binomial distribution as probability of success changes with each draw
• Applications include quality control sampling and lottery draws
• Mean of Hypergeometric distribution calculated as $E(X) = n\frac{K}{N}$
• Variance determined by $Var(X) = n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$

Distribution Functions

Probability Mass Function (PMF)

• Probability Mass Function defines probability of discrete random variable taking specific value
• For discrete random variable X, PMF given by $P(X=x)$ for each possible value x
• PMF must satisfy two conditions: $P(X=x) \geq 0$ for all x, and $\sum_x P(X=x) = 1$
• Represents probability distribution for discrete random variables
• Used to calculate probabilities of specific outcomes or ranges of outcomes
• Visualized as bar graph with heights representing probabilities

Cumulative Distribution Function (CDF)

• Cumulative Distribution Function gives probability of random variable being less than or equal to specific value
• For discrete random variable X, CDF defined as $F_X(x) = P(X \leq x)$
• CDF calculated by summing PMF values: $F_X(x) = \sum_{k \leq x} P(X=k)$
• Properties of CDF include monotonically increasing and right-continuous
• Limits of CDF approach 0 as x approaches negative infinity and 1 as x approaches positive infinity
• Used to find probabilities of intervals and percentiles of distribution
• Relationship between PMF and CDF: $P(X=x) = F_X(x) - F_X(x^-)$ where x^- is value just below x

Distribution Properties

Expected Value and Its Applications

• Expected value represents average outcome of random variable over many trials
• For discrete random variable X, expected value calculated as $E(X) = \sum_x x P(X=x)$
• Provides measure of central tendency for probability distribution
• Linearity of expectation allows $E(aX + b) = aE(X) + b$ for constants a and b
• Used in decision making, risk assessment, and financial modeling
• Applications include calculating average winnings in games of chance
• Expected value of function g(X) given by $E[g(X)] = \sum_x g(x) P(X=x)$

Variance and Standard Deviation

• Variance measures spread or dispersion of random variable around its expected value
• For discrete random variable X, variance calculated as $Var(X) = E[(X - E(X))^2] = \sum_x (x - E(X))^2 P(X=x)$
• Alternative formula for variance: $Var(X) = E(X^2) - [E(X)]^2$
• Standard deviation defined as square root of variance: $\sigma = \sqrt{Var(X)}$
• Properties of variance include $Var(aX + b) = a^2Var(X)$ for constants a and b
• Used to assess risk and uncertainty in various fields (finance, engineering, sciences)
• Chebyshev's inequality relates variance to probability of deviations from mean
• Coefficient of variation given by ratio of standard deviation to mean: $CV = \frac{\sigma}{\mu}$