Topic: Statistics (for undergraduates and junior college)
Students are commonly asked what assumptions are needed for commonly-used statistical distributions such as the binomial or exponential distribution. Here is a brief summary.
Binomial and Geometric Distributions:
- Each trial is independent of the others.
- Each trial results two outcomes — success or failure, whose probabilities are constant.
- If the random variable X denotes the number of successful trials out of a random sample of n trials, and the probability of success is p, then X has a Binomial distribution B(n,p) with mean np and variance np(1-p).
- If the random variable Y denotes the number of trials needed up to and including the first successful trial, and the probability of success is p, then Y has a Geometric distribution Geo(p) with mean 1/p.
Poisson and Exponential Distributions:
- Events occur randomly and independently of each other.
- The average number of occurrences is constant.
- If the random variable X denotes the number of occurrences within a specified timeframe, and the mean number of occurrences for that timeframe is u, then X has a Poisson distribution Po(u).
- If the random variable Y denotes the time between consecutive occurrences, then Y has an Exponential distribution exp(u), whose mean is 1/u.