📉Intro to Business Statistics Unit 4 – Discrete Random Variables

What are Discrete Random Variables?

• Discrete random variables take on a countable number of distinct values
• Typically represented by uppercase letters (X, Y, Z)
• Outcomes are often integers or whole numbers (0, 1, 2, 3...)
• Probability of each outcome is between 0 and 1, inclusive
  • Sum of all probabilities must equal 1
• Examples include:
  • Number of defective items in a batch (0, 1, 2...)
  • Number of customers arriving at a store per hour (0, 1, 2...)
• Contrast with continuous random variables, which can take on any value within a range
• Discrete random variables are the foundation for many probability distributions

Probability Mass Functions (PMF)

• PMF defines the probability of each possible outcome for a discrete random variable
• Denoted as P(X = x), where X is the random variable and x is a specific outcome
• Must satisfy two conditions:
  • P(X = x) ≥ 0 for all x
  • Sum of P(X = x) over all possible x values equals 1
• Can be represented as a table, graph, or formula
• Example: PMF for the number of heads in two coin flips (0, 1, or 2)
  • P(X = 0) = 0.25, P(X = 1) = 0.5, P(X = 2) = 0.25
• Helps calculate probabilities of specific outcomes or ranges of outcomes
• Foundation for calculating expected value and variance

Expected Value and Variance

• Expected value (mean) is the average value of a discrete random variable over many trials
  • Denoted as E(X) or μ
  • Calculated as the sum of each outcome multiplied by its probability: $E(X) = \sum x \cdot P(X = x)$
• Variance measures the spread or dispersion of a discrete random variable around its expected value
  • Denoted as Var(X) or σ^2
  • Calculated as the sum of squared deviations from the mean, weighted by their probabilities: $Var(X) = \sum (x - \mu)^2 \cdot P(X = x)$
  • Standard deviation is the square root of variance: $\sigma = \sqrt{Var(X)}$
• Both expected value and variance provide insights into the behavior of a discrete random variable
• Used to compare different distributions and make informed decisions

Common Discrete Distributions

• Bernoulli distribution: models a single trial with two possible outcomes (success or failure)
  • P(X = 1) = p, P(X = 0) = 1 - p, where p is the probability of success
• Binomial distribution: models the number of successes in a fixed number of independent Bernoulli trials
  • Denoted as X ~ Bin(n, p), where n is the number of trials and p is the probability of success
  • PMF: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, where k is the number of successes
• Poisson distribution: models the number of events occurring in a fixed interval of time or space
  • Denoted as X ~ Poisson(λ), where λ is the average rate of events
  • PMF: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$, where k is the number of events
• Geometric distribution: models the number of trials until the first success occurs
  • Denoted as X ~ Geo(p), where p is the probability of success
  • PMF: $P(X = k) = (1-p)^{k-1}p$, where k is the number of trials
• Hypergeometric distribution: models the number of successes in a fixed number of draws without replacement from a finite population
  • Denoted as X ~ HGeom(N, K, n), where N is the population size, K is the number of successes in the population, and n is the number of draws
  • PMF: $P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$, where k is the number of successes in the sample

Solving Problems with Discrete Random Variables

• Identify the appropriate discrete random variable and distribution for the given problem
• Determine the parameters of the distribution (e.g., p for Bernoulli, n and p for Binomial)
• Write the PMF or use built-in functions in software (Excel, R, Python) to calculate probabilities
• Calculate the expected value and variance using the formulas or built-in functions
• Interpret the results in the context of the problem
• Example: A fair die is rolled 3 times. Let X be the number of times a 6 appears. Find P(X = 2) and E(X).
  • X ~ Bin(3, 1/6) since there are 3 trials and the probability of success (rolling a 6) is 1/6
  • P(X = 2) = $\binom{3}{2} (1/6)^2 (5/6)^1 \approx 0.0463$
  • E(X) = np = 3 × (1/6) = 0.5
• Adapt problem-solving strategies to different scenarios and distributions

Real-World Applications in Business

• Quality control: model the number of defective items in a batch using the Binomial or Poisson distribution
• Customer service: model the number of customer complaints per day using the Poisson distribution
• Inventory management: model the demand for a product using the Poisson or Geometric distribution
• Marketing: model the number of people who respond to a promotional offer using the Binomial distribution
• Finance: model the number of defaults in a loan portfolio using the Binomial distribution
• Project management: model the number of tasks completed on time using the Binomial distribution
• Supply chain: model the number of supplier deliveries that arrive late using the Poisson distribution
• Applying discrete random variables in real-world scenarios helps make informed business decisions and manage risks

Key Formulas and Concepts to Remember

• PMF: P(X = x), defines the probability of each outcome for a discrete random variable
• Expected value: $E(X) = \sum x \cdot P(X = x)$, the average value of a discrete random variable
• Variance: $Var(X) = \sum (x - \mu)^2 \cdot P(X = x)$, measures the spread of a discrete random variable
• Standard deviation: $\sigma = \sqrt{Var(X)}$, square root of variance
• Bernoulli distribution: models a single trial with two possible outcomes
• Binomial distribution: models the number of successes in a fixed number of independent Bernoulli trials
  • PMF: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
• Poisson distribution: models the number of events occurring in a fixed interval of time or space
  • PMF: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
• Geometric distribution: models the number of trials until the first success occurs
  • PMF: $P(X = k) = (1-p)^{k-1}p$
• Hypergeometric distribution: models the number of successes in a fixed number of draws without replacement from a finite population
  • PMF: $P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$

Practice Problems and Solutions

1. A fair coin is tossed 5 times. Let X be the number of heads. Find P(X ≥ 3) and E(X).

   • X ~ Bin(5, 0.5)
   • P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) ≈ 0.5
   • E(X) = np = 5 × 0.5 = 2.5

2. A call center receives an average of 4 calls per minute. Find the probability of receiving exactly 3 calls in a 2-minute period.

   • X ~ Poisson(4 × 2 = 8)
   • P(X = 3) = $\frac{e^{-8}8^3}{3!} \approx 0.0498$

3. A machine produces defective items with a probability of 0.02. Find the probability of observing at most 2 defective items in a batch of 100.

   • X ~ Bin(100, 0.02)
   • P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.9999

4. The probability of a customer making a purchase is 0.3. Find the probability that the 4th customer is the first to make a purchase.

   • X ~ Geo(0.3)
   • P(X = 4) = $(1-0.3)^{3}0.3 \approx 0.1029$

5. A box contains 20 red marbles and 30 blue marbles. If 10 marbles are drawn at random without replacement, find the probability of drawing exactly 4 red marbles.

   • X ~ HGeom(50, 20, 10)
   • P(X = 4) = $\frac{\binom{20}{4}\binom{30}{6}}{\binom{50}{10}} \approx 0.2517$