Category: Probability concepts. These pages discuss some of the practical and theoretical considerations concerning probability. Articles are arranged by date with the most recent entries at the top. You can find the theme and closely related categories, definitions, and other resources at the bottom of this page.
Stats: Calculating probabilities involving correlated normal variables (June 4, 2007). Someone on EDSTAT-L asked about a problem involving differences of independent normal random variables. I am simplifying the problem a bit, but it essentially asked a question that was comparable to the following: Suppose you have three independent standard normal random variables: A, B, and C. What is the probability that A is smaller than B by one or more units and A is also smaller than B by one or more units.
Stats: Formulas for cumulative Poisson and binomial probabilities (February 19, 2007). I am updating some material about Poisson regression and noticed that some of the tests and confidence intervals rely on a percentile from a Chi-squared distribution or a gamma distribution. In previous work on binomial confidence intervals, I had noticed the use of a beta distribution and an F distribution. It seems odd to apply percentiles from continuous distributions for confidence intervals involving counting, but the formulas do indeed work. There are well known relationships for the cumulative distributions of the Poisson and binomial distributions that lead to these formulas.
Stats: Extreme value distribution (January 9, 2006). I got an interesting question about an application in information theory of a statistical distribution called the Type I extreme value distribution. This distribution, also known as the Gumbel distribution, is useful for modeling the maximum or minimum of a large number of variables.
Stats: Expected value and moments (July 29, 2005). Someone asked me what a statistical moment is. That's a rather technical term and is not needed except in rather theoretical and mathematical discussions. But it is still worth defining.
Stats: Geometric distribution (May 16, 2005). Someone asked me about a game where A, B, and C toss a coin in order until someone gets a heads on their coin flip. What are the probabilities that A will win? B will win? C will win?
Stats: Testing multinomial proportions (November 9, 2004). I received an email inquiry about a problem that seems simple enough, but which just doesn't seem to have an easy answer. This person gave the following hypothetical data: Suppose in a sample of 100 people, 21 have blue eyes and 23 have green eyes. Can you test the hypothesis that the proportion of blue eyes is equal to the proportion of green eyes? This is not a two sample binomial problem and it is not a one sample binomial problem either. The only way you can properly analyze this data is to treat it as a single multinomial sample.
Theme and closely related categories:
- Stats: What is a binomial distribution?
- Stats: What is a binomial mean?
- Stats: What is a binomial probability?
- Stats: What is entropy?
- Stats: What is independence?
- Stats: What is a normal distribution?
- Stats: What is a normal probability?
- Stats: What are odds?
- Stats: What is a Poisson distribution?
- [coming soon]
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This webpage was written by Steve Simon on 2007-09-21, edited by Steve Simon, and was last modified on 2008-07-08. Send feedback to ssimon at cmh dot edu or click on the email link at the top of the page.