Probability Distributions: Binomial, Normal, and Poisson
Probability distributions describe how likely outcomes are in a random experiment. The binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes. The normal distribution, or bell curve, represents continuous data that cluster around a mean, showing natural variability. The Poisson distribution describes the probability of a given number of events happening in a fixed interval of time or space, often used for rare events.
Probability Distributions: Binomial, Normal, and Poisson
Probability distributions describe how likely outcomes are in a random experiment. The binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes. The normal distribution, or bell curve, represents continuous data that cluster around a mean, showing natural variability. The Poisson distribution describes the probability of a given number of events happening in a fixed interval of time or space, often used for rare events.
💡 Key Takeaways
- Decide when to use Binomial, Normal, or Poisson distributions and understand their basic assumptions.
- Identify key parameters: Binomial (n, p); Normal (μ, σ); Poisson (λ) and their means and variances.
- Compute probabilities with the appropriate function (PMF for Binomial/Poisson, PDF for Normal) and apply common approximations (Normal approx to Binomial; Poisson approx to Binomial).
- Interpret results in real problems by distinguishing discrete vs continuous data and what counts of successes or rare events imply.
❓ Frequently Asked Questions
What is a probability distribution?
A rule that assigns probabilities to outcomes of a random experiment, describing how likely each result is.
What is the Binomial distribution and its key parameters?
Models the number of successes in n independent trials with the same probability p of success. P(X=k) = C(n,k) p^k (1−p)^{n−k}; mean = np, variance = np(1−p).
What is the Normal distribution and when is it used?
A continuous, symmetric bell-shaped distribution with mean μ and standard deviation σ. Describes many natural data patterns and follows the 68-95-99.7 rules.
What is the Poisson distribution and when is it used?
Models the count of independent events in a fixed interval with rate λ. P(X=k) = e^{−λ} λ^k / k!; mean = variance = λ.
When should you use a normal approximation to a binomial?
When n is large and p is not too close to 0 or 1 (commonly np ≥ 5 and n(1−p) ≥ 5), often with a continuity correction.