Types of Probability Distributions to Know for AP Statistics

Understanding different probability distributions is key in statistics. Each type, like Normal, Binomial, and Poisson, helps model real-world scenarios, guiding decisions in fields like biostatistics and research. These distributions form the backbone of statistical analysis and inference.

1. Normal Distribution

   • Symmetrical, bell-shaped curve characterized by its mean (μ) and standard deviation (σ).
   • Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (Empirical Rule).
   • Central to many statistical methods, including hypothesis testing and confidence intervals.

2. Binomial Distribution

   • Models the number of successes in a fixed number of independent Bernoulli trials (n), each with the same probability of success (p).
   • Defined by two parameters: n (number of trials) and p (probability of success).
   • Useful for scenarios with yes/no outcomes, such as flipping a coin or passing a test.

3. Poisson Distribution

   • Describes the number of events occurring in a fixed interval of time or space, given a known average rate (λ) and independence of events.
   • Suitable for rare events, such as the number of phone calls received at a call center in an hour.
   • The mean and variance of a Poisson distribution are both equal to λ.

4. Uniform Distribution

   • All outcomes are equally likely within a defined range, characterized by minimum (a) and maximum (b) values.
   • Can be discrete (finite number of outcomes) or continuous (infinite outcomes within an interval).
   • The probability density function is constant across the interval [a, b].

5. Exponential Distribution

   • Models the time until an event occurs, such as the time between arrivals in a queue.
   • Defined by a single parameter (λ), which is the rate of occurrence.
   • Memoryless property: the probability of an event occurring in the next time interval is independent of how much time has already elapsed.

6. Chi-Square Distribution

   • Used primarily in hypothesis testing and constructing confidence intervals for variance and goodness-of-fit tests.
   • Defined by degrees of freedom (df), which typically correspond to the number of categories minus one.
   • The distribution is positively skewed and approaches a normal distribution as df increases.

7. Student's t-Distribution

   • Similar to the normal distribution but with heavier tails, making it more suitable for small sample sizes.
   • Defined by degrees of freedom (df), which affect the shape of the distribution.
   • Used in hypothesis testing and constructing confidence intervals when the population standard deviation is unknown.

8. F-Distribution

   • Used primarily in analysis of variance (ANOVA) and comparing variances between two populations.
   • Defined by two sets of degrees of freedom: one for the numerator and one for the denominator.
   • Positively skewed and approaches a normal distribution as the degrees of freedom increase.

9. Bernoulli Distribution

   • Represents a single trial with two possible outcomes: success (1) or failure (0).
   • Defined by a single parameter (p), the probability of success.
   • The foundation for the binomial distribution, as it models the simplest case of a binomial trial.

10. Geometric Distribution

    • Models the number of trials needed to achieve the first success in a series of independent Bernoulli trials.
    • Defined by the probability of success (p) on each trial.
    • Memoryless property: the probability of success in future trials does not depend on past trials.