Module 4: Probability Distributions - Discrete
Supporting Lectures:
EGN3443 Module 4 - Discrete Probability Distributions
1. Random Variables and Probability Mass Functions (PMF)
Definition
A random variable is a function that assigns a numerical value to each possible outcome of a random experiment. For discrete random variables, a Probability Mass Function (PMF) specifies the probability of each possible value the random variable can take.
Key Characteristics
- Denoted typically as P(X = x)
- Always non-negative
- Sum of all probabilities equals 1
Example
Consider a die roll (X):
- X can take values 1, 2, 3, 4, 5, or 6
- PMF: P(X = k) = 1/6 for k = 1, 2, 3, 4, 5, 6
- Each outcome has an equal probability of 1/6
Web Reference:
2. Expected Value and Variance
Expected Value (Mean)
The expected value represents the average outcome of a random variable over many trials.
Formula: E(X) = Σ [x * P(X = x)]
Variance
Variance measures the spread of a probability distribution.
Formula: Var(X) = E[(X - E(X))²] = Σ [(x - E(X))² * P(X = x)]
Calculation Example
Die Roll Scenario:
- E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5
- Variance calculation involves more complex steps
Web Reference:
3. Bernoulli and Binomial Distributions
Bernoulli Distribution
A discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability (1-p).
Parameters:
- p: Probability of success
- n: Number of trials
PMF: P(X = k) = p^k * (1-p)^(1-k), where k ∈ {0, 1}
The sum of all the probabilities of all the possible outcomes will 1.0
A stem plot is a good method of visualizing the PMF.
Binomial Distribution
Represents the number of successes in n independent Bernoulli trials.
Formula: P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
- C(n,k) is the binomial coefficient
- k is the number of successes
- n is the total number of trials
- p is the probability of success
Example Calculation
Coin Flips:
- n = 10 flips
- p = 0.5 (fair coin)
- Probability of exactly 7 heads: P(X = 7) = C(10,7) * (0.5)^7 * (0.5)^3 ≈ 0.117
Web Reference: https://en.wikipedia.org/wiki/Binomial_distribution
4. Poisson Distribution
Definition
Models the number of events occurring in a fixed interval of time or space, given a constant mean rate.
Parameters:
- λ (lambda): Average number of events per interval
- k: Actual number of events
PMF: P(X = k) = (e^-λ * λ^k) / k!
Applications
- Modeling rare events
- Customer arrivals
- Radioactive decay
- Network packet arrivals
Example Calculation
Call Center Scenario:
- Average 5 calls per hour (λ = 5)
- Probability of exactly 3 calls in an hour: P(X = 3) = (e^-5 * 5^3) / 3! ≈ 0.1403
Web Reference: https://en.wikipedia.org/wiki/Poisson_distribution
5. Hypergeometric Distribution
Definition
Probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes.
Parameters:
- N: Total population size
- K: Number of success states in the population
- n: Number of draws
- k: Number of observed successes
PMF: P(X = k) = [C(K,k) * C(N-K, n-k)] / C(N, n)
Example
Quality Control:
- 100 items (N), 20 defective (K)
- Inspect 10 items (n)
- Probability of finding exactly 3 defective items (k)
Web Reference: https://en.wikipedia.org/wiki/Hypergeometric_distribution
Recommended Study Resources
- OpenStax Probability and Statistics
- MIT OpenCourseWare - Probability
- EdX - Probability Courses
Practice Suggestions
- Solve problems from textbooks
- Use online probability calculators
- Create simulation experiments
- Practice deriving PMFs for different scenarios
Common Pitfalls to Avoid
- Confusing discrete and continuous distributions
- Misapplying probability formulas
- Overlooking independence assumptions
- Computational errors in complex calculations