Intro to Probability Unit 8 Review: Bernoulli to Poisson

Key Concepts

• Discrete probability distributions assign probabilities to discrete random variables
• Probability mass function (PMF) defines the probability of each possible value of a discrete random variable
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
• Expected value represents the average value of a random variable over many trials
• Variance measures the spread or dispersion of a random variable around its expected value
  • Calculated as the average squared deviation from the mean
• Independent and identically distributed (i.i.d.) random variables have the same probability distribution and are mutually independent
• Memoryless property states that the probability of an event occurring is independent of the past history of the process

Bernoulli Distribution

• Models a single trial with two possible outcomes: success (probability $p$) and failure (probability $1-p$)
• Random variable $X$ follows a Bernoulli distribution with parameter $p$, denoted as $X \sim \text{Bern}(p)$
• Probability mass function: $P(X = x) = p^x (1-p)^{1-x}$ for $x \in \{0, 1\}$
  • $P(X = 1) = p$ and $P(X = 0) = 1-p$
• Expected value: $E(X) = p$
• Variance: $\text{Var}(X) = p(1-p)$
• Used to model binary outcomes (coin flips, defective/non-defective items, pass/fail exams)

Binomial Distribution

• Models the number of successes in a fixed number of independent Bernoulli trials with constant success probability
• Random variable $X$ follows a binomial distribution with parameters $n$ and $p$, denoted as $X \sim \text{Bin}(n, p)$
• Probability mass function: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ for $k = 0, 1, \ldots, n$
  • $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials
• Expected value: $E(X) = np$
• Variance: $\text{Var}(X) = np(1-p)$
• Models the number of successes in a fixed number of trials (number of heads in 10 coin flips, number of defective items in a batch)

Geometric Distribution

• Models the number of trials until the first success in a sequence of independent Bernoulli trials with constant success probability
• Random variable $X$ follows a geometric distribution with parameter $p$, denoted as $X \sim \text{Geom}(p)$
• Probability mass function: $P(X = k) = (1-p)^{k-1}p$ for $k = 1, 2, \ldots$
• Expected value: $E(X) = \frac{1}{p}$
• Variance: $\text{Var}(X) = \frac{1-p}{p^2}$
• Memoryless property: $P(X > m+n | X > m) = P(X > n)$ for any non-negative integers $m$ and $n$
• Models waiting times until the first success (number of coin flips until the first head, number of quality checks until the first defective item)

Negative Binomial Distribution

• Generalizes the geometric distribution to model the number of trials until the $r$-th success in a sequence of independent Bernoulli trials with constant success probability
• Random variable $X$ follows a negative binomial distribution with parameters $r$ and $p$, denoted as $X \sim \text{NB}(r, p)$
• Probability mass function: $P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}$ for $k = r, r+1, \ldots$
• Expected value: $E(X) = \frac{r}{p}$
• Variance: $\text{Var}(X) = \frac{r(1-p)}{p^2}$
• Models the number of trials until a fixed number of successes (number of coin flips until the 5th head, number of job interviews until 3 offers)

Poisson Distribution

• Models the number of rare events occurring in a fixed interval of time or space, given a known average rate of occurrence
• Random variable $X$ follows a Poisson distribution with parameter $\lambda$, denoted as $X \sim \text{Pois}(\lambda)$
• Probability mass function: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$ for $k = 0, 1, 2, \ldots$
• Expected value: $E(X) = \lambda$
• Variance: $\text{Var}(X) = \lambda$
• Poisson process: Events occur independently and at a constant average rate
  • Inter-arrival times between events follow an exponential distribution with rate $\lambda$
• Approximates binomial distribution when $n$ is large and $p$ is small, such that $np = \lambda$
• Models rare events (number of car accidents per day, number of typos per page, number of customers arriving per hour)

Applications and Examples

• Quality control: Binomial distribution to model the number of defective items in a batch, geometric distribution to model the number of inspections until a defective item is found
• Genetics: Binomial distribution to model the number of individuals with a specific genotype in a population
• Finance: Geometric distribution to model the number of days until a stock price exceeds a certain threshold
• Queueing theory: Poisson distribution to model the number of customers arriving at a service counter per hour
• Reliability engineering: Negative binomial distribution to model the number of component failures until a system breaks down
• Epidemiology: Poisson distribution to model the number of new cases of a rare disease in a population per year
• Telecommunications: Poisson distribution to model the number of phone calls arriving at a call center per minute

Common Pitfalls and Tips

• Ensure that the assumptions of each distribution are met before applying them to a problem
  • Independent trials, constant success probability, and fixed number of trials for binomial distribution
  • Rare events occurring independently and at a constant average rate for Poisson distribution
• Be careful when using the Poisson approximation to the binomial distribution, as it is only valid when $n$ is large and $p$ is small
• Remember that the geometric and negative binomial distributions start counting from the first trial, while the binomial distribution counts the total number of successes in a fixed number of trials
• Use the memoryless property of the geometric distribution to simplify calculations when appropriate
• When working with Poisson distribution, make sure to consider the units of the parameter $\lambda$ (e.g., events per unit time or space)
• Double-check the formulas for PMF, expected value, and variance, as they differ for each distribution
• Practice solving problems using various approaches, such as using the PMF, CDF, or moment-generating functions, to gain a deeper understanding of the distributions