Discrete probability distributions are the backbone of statistical analysis for events with distinct outcomes. They help us understand and predict the likelihood of specific results in scenarios like coin flips, product defects, or customer arrivals.

This section dives into key discrete distributions like binomial and Poisson. We'll explore their characteristics, how to calculate probabilities, and their practical applications in engineering and everyday life.

Discrete Random Variables

Definition and Characteristics

• A discrete random variable is a function that assigns a probability to each possible outcome in a sample space of a random experiment
  • It can only take on a finite or countably infinite number of distinct values (integers, whole numbers)
• The sum of the probabilities for all possible values of a discrete random variable must equal 1
  • This ensures that the total probability of all outcomes is 100%
• The probability mass function (PMF) gives the probability that a discrete random variable is exactly equal to a specific value
  • PMF is denoted as P(X = x), where X is the random variable and x is a specific value
• The cumulative distribution function (CDF) is the probability that a random variable X takes a value less than or equal to x, for every value x
  • CDF is denoted as F(x) = P(X ≤ x)

Calculating Probabilities

• To find the probability of a specific value of a discrete random variable, use the probability mass function (PMF) to identify the probability associated with that value
  • Example: If P(X = 3) = 0.2, then the probability of the random variable X taking the value 3 is 0.2 or 20%
• To find the probability that a discrete random variable falls within a range of values, add the probabilities of each individual value within the range using the PMF
  • Example: To find P(1 ≤ X ≤ 4), calculate P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
• To find the probability that a discrete random variable is less than or equal to a specific value, use the cumulative distribution function (CDF) to identify the cumulative probability up to and including that value
  • Example: If F(5) = 0.8, then P(X ≤ 5) = 0.8 or 80%

Probability Distributions for Discrete Variables

Common Discrete Probability Distributions

• The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success
  • Its PMF is given by the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
    • n is the number of trials
    • k is the number of successes
    • p is the probability of success on each trial
    • C(n, k) is the number of ways to choose k items from n items (binomial coefficient)
• The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event
  • Its PMF is given by the formula P(X = k) = (λ^k e^-λ) / k!
    • λ is the average number of events per interval
    • e is Euler's number (approximately 2.71828)
    • k! is the factorial of k
• The geometric distribution models the number of Bernoulli trials needed to get the first success
  • Its PMF is given by the formula P(X = k) = (1-p)^(k-1) p
    • k is the number of trials until the first success
    • p is the probability of success on each trial

Applying Discrete Probability Distributions

• Binomial distribution example: The probability of getting exactly 3 heads in 5 coin tosses, with a fair coin (p = 0.5)
  • P(X = 3) = C(5, 3) * 0.5^3 * (1-0.5)^(5-3) ≈ 0.3125 or 31.25%
• Poisson distribution example: The probability of 2 customers arriving in a 10-minute interval at a store where the average arrival rate is 3 customers per hour (λ = 0.5 customers per 10 minutes)
  • P(X = 2) = (0.5^2 e^-0.5) / 2! ≈ 0.0758 or 7.58%
• Geometric distribution example: The probability of needing 4 attempts to get the first success in a game with a 20% chance of winning each time (p = 0.2)
  • P(X = 4) = (1-0.2)^(4-1) 0.2 ≈ 0.1024 or 10.24%

Binomial and Poisson Distributions

Binomial Distribution

• The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success
  • The trials must be independent, meaning the outcome of one trial does not affect the outcome of another
  • The probability of success (p) must remain constant across all trials
• The PMF of the binomial distribution is given by the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
  • n is the number of trials
  • k is the number of successes
  • p is the probability of success on each trial
  • C(n, k) is the binomial coefficient, calculated as n! / (k! (n-k)!)
• Examples of situations that can be modeled by the binomial distribution include:
  • The number of defective items in a batch of products
  • The number of successful free throws in a basketball game
  • The number of correct answers on a multiple-choice exam

Poisson Distribution

• The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event
  • The events must be independent of each other
  • The average rate of occurrence (λ) must remain constant over the interval
• The PMF of the Poisson distribution is given by the formula P(X = k) = (λ^k e^-λ) / k!
  • λ is the average number of events per interval
  • e is Euler's number (approximately 2.71828)
  • k! is the factorial of k
• Examples of situations that can be modeled by the Poisson distribution include:
  • The number of phone calls received by a call center per hour
  • The number of defects per unit area in a fabric
  • The number of accidents at an intersection per month

Expected Value and Variance of Discrete Distributions

Expected Value (Mean)

• The expected value (or mean) of a discrete random variable is the sum of the products of each possible value and its probability
  • It represents the average value of the random variable over a large number of trials
• The expected value is calculated using the formula E(X) = Σ[x P(X=x)]
  • x represents each possible value of the random variable X
  • P(X=x) is the probability of X taking on the value x
• For the binomial distribution with parameters n and p, the expected value is E(X) = np
  • Example: If n = 10 and p = 0.4, then E(X) = 10 0.4 = 4
• For the Poisson distribution with parameter λ, the expected value is equal to λ
  • Example: If λ = 3.5, then E(X) = 3.5

Variance and Standard Deviation

• The variance of a discrete random variable measures the average squared deviation from the expected value
  • It quantifies the spread or dispersion of the random variable's values around the mean
• The variance is calculated using the formula Var(X) = E(X^2) - [E(X)]^2
  • E(X^2) is the expected value of the squared random variable
  • [E(X)]^2 is the square of the expected value
• The standard deviation is the square root of the variance and measures the average distance of the values from the mean
  • It is denoted as σ(X) or SD(X) and calculated as σ(X) = √(Var(X))
• For the binomial distribution with parameters n and p, the variance is Var(X) = np(1-p)
  • Example: If n = 10 and p = 0.4, then Var(X) = 10 * 0.4 * (1-0.4) = 2.4
• For the Poisson distribution with parameter λ, the variance is equal to λ
  • Example: If λ = 3.5, then Var(X) = 3.5