📈Preparatory Statistics Unit 7 – Discrete Variables & Probability Distributions

Key Concepts

• Discrete variables take on a finite or countably infinite number of distinct values
• Probability is the likelihood of an event occurring, expressed as a number between 0 and 1
  • 0 indicates an impossible event, while 1 represents a certain event
• Probability distributions describe the likelihood of each possible outcome for a discrete variable
• Expected value is the average value of a discrete random variable over many trials, calculated as the sum of each outcome multiplied by its probability
• Variance and standard deviation measure the spread or dispersion of a probability distribution
  • Variance is the average squared deviation from the mean, while standard deviation is the square root of variance
• Independence means that the occurrence of one event does not affect the probability of another event
• Mutually exclusive events cannot occur simultaneously, and their probabilities sum to 1

Types of Discrete Variables

• Bernoulli variables have only two possible outcomes, typically labeled as success (1) or failure (0)
  • Examples include flipping a coin (heads or tails) or a yes/no survey question
• Binomial variables count the number of successes in a fixed number of independent Bernoulli trials
  • Parameters are the number of trials ($n$) and the probability of success ($p$)
• Poisson variables count the number of events occurring in a fixed interval of time or space
  • The parameter is the average rate of occurrence ($\lambda$)
• Geometric variables count the number of trials until the first success in a series of independent Bernoulli trials
  • The parameter is the probability of success ($p$)
• Hypergeometric variables count the number of successes in a fixed number of trials without replacement from a finite population
  • Parameters include the population size, the number of successes in the population, and the number of trials

Probability Basics

• The probability of an event $A$ is denoted as $P(A)$
• The complement of an event $A$, denoted as $A'$ or $\bar{A}$, is the event that $A$ does not occur
  • $P(A') = 1 - P(A)$
• The union of two events $A$ and $B$, denoted as $A \cup B$, is the event that either $A$ or $B$ occurs
  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• The intersection of two events $A$ and $B$, denoted as $A \cap B$, is the event that both $A$ and $B$ occur
  • For independent events, $P(A \cap B) = P(A) \times P(B)$
• Conditional probability is the probability of an event $A$ occurring given that event $B$ has occurred, denoted as $P(A|B)$
  • $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Bayes' theorem relates conditional probabilities and can be used to update probabilities based on new information
  • $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$

Discrete Probability Distributions

• A probability mass function (PMF) gives the probability of each possible outcome for a discrete random variable
  • $f(x) = P(X = x)$, where $X$ is the random variable and $x$ is a specific value
• The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
  • $F(x) = P(X \leq x) = \sum_{y \leq x} f(y)$
• The expected value (mean) of a discrete random variable $X$ is $E(X) = \sum_{x} x \times f(x)$
• The variance of a discrete random variable $X$ is $Var(X) = E((X - \mu)^2) = \sum_{x} (x - \mu)^2 \times f(x)$, where $\mu = E(X)$
  • The standard deviation is the square root of the variance, $\sigma = \sqrt{Var(X)}$
• The moment-generating function (MGF) is a tool for finding moments of a distribution, such as the mean and variance
  • $M_X(t) = E(e^{tX}) = \sum_{x} e^{tx} \times f(x)$

Common Discrete Distributions

• Bernoulli distribution: $P(X = 1) = p$, $P(X = 0) = 1 - p$
  • $E(X) = p$, $Var(X) = p(1 - p)$
• Binomial distribution: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$
  • $E(X) = np$, $Var(X) = np(1 - p)$
• Poisson distribution: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$
  • $E(X) = \lambda$, $Var(X) = \lambda$
• Geometric distribution: $P(X = k) = (1 - p)^{k-1} p$
  • $E(X) = \frac{1}{p}$, $Var(X) = \frac{1 - p}{p^2}$
• Hypergeometric distribution: $P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$
  • $E(X) = n \frac{K}{N}$, $Var(X) = n \frac{K}{N} (1 - \frac{K}{N}) \frac{N-n}{N-1}$

Calculating Probabilities

• To find the probability of a specific outcome, use the PMF for the appropriate distribution
  • Substitute the given parameters and the desired outcome into the formula
• To find the probability of a range of outcomes, sum the probabilities of each individual outcome in the range
  • Alternatively, use the CDF to find $P(a \leq X \leq b) = F(b) - F(a)$
• For independent events, multiply the probabilities of each event to find the probability of their intersection
• For mutually exclusive events, add the probabilities of each event to find the probability of their union
• Use conditional probability and Bayes' theorem when the probability of an event depends on the occurrence of another event

Applications in Real-World Scenarios

• Quality control: Model the number of defective items in a batch using a binomial distribution
  • Determine the probability of accepting or rejecting a batch based on the number of defects found in a sample
• Insurance claims: Use a Poisson distribution to model the number of claims filed within a given time period
  • Calculate the probability of exceeding a certain number of claims to set premiums and reserves
• Genetics: Apply the hypergeometric distribution to calculate probabilities in population genetics
  • For example, find the probability of observing a specific number of individuals with a rare allele in a sample from a larger population
• Marketing: Employ the geometric distribution to model the number of ads a customer sees before making a purchase
  • Estimate the expected number of ads needed to generate a sale and optimize ad placement strategies
• Reliability engineering: Utilize the negative binomial distribution to model the number of failures before a specified number of successes
  • Assess the reliability of systems or components and plan maintenance schedules accordingly

Practice Problems and Examples

• A fair coin is flipped 10 times. What is the probability of observing exactly 7 heads?
  • $P(X = 7) = \binom{10}{7} (0.5)^7 (0.5)^3 \approx 0.1172$
• The average number of customers entering a store per hour is 30. What is the probability that more than 35 customers enter the store in a given hour?
  • Using the Poisson distribution with $\lambda = 30$, $P(X > 35) = 1 - P(X \leq 35) \approx 0.1185$
• In a batch of 100 items, 10 are known to be defective. If 20 items are randomly selected without replacement, what is the probability that exactly 3 of them are defective?
  • Using the hypergeometric distribution, $P(X = 3) = \frac{\binom{10}{3} \binom{90}{17}}{\binom{100}{20}} \approx 0.1954$
• A machine produces defective items with a probability of 0.02. What is the probability that the 5th defective item is produced on the 200th item?
  • Using the negative binomial distribution, $P(X = 200) = \binom{199}{4} (0.02)^5 (0.98)^{195} \approx 0.0190$
• Two fair dice are rolled. What is the probability that the sum of the numbers rolled is either 7 or 11?
  • $P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}$ and $P(\text{sum} = 11) = \frac{2}{36} = \frac{1}{18}$
  • Since these events are mutually exclusive, $P(\text{sum} = 7 \text{ or } 11) = \frac{1}{6} + \frac{1}{18} = \frac{2}{9}$