Diagram of probability distribution relationships
A continuous random variable X is said to have an Exponential(λ) distribution if it has .. an increment is the difference in the process at two times, say s and t. For sdistribution of Sn is Gamma with parameters n and λ. Before. Relation to Other Distributions Beta(1, 1) = Uniform(0, 1). 3 Bernoulli . which for integer r agrees with the standard definition. Also. (r + x − 1 . The relationship between the Exponential(λ) and Gamma(1,λ) distributions gives. and we can see how the standard gamma depends on α. See Figure gamma with α=1 is the exponential distribution (defined on p even though it is a So we have a sort of box of links: Geometric. Exponential. Neg Binomial. Gamma.
The difference between a hypergeometric distribution and a binomial distribution is the difference between sampling without replacement and sampling with replacement. As the population size increases relative to the sample size, the difference becomes negligible.
The relationship is simpler in terms of failure probabilities: For more information, see Poisson approximation to binomial. The sum of n Bernoulli p random variables is a binomial n, p random variable.
For more information, see normal approximation to Poisson. If X is a binomial n, p random variable and Y is a normal random variable with the same mean and variance as X, i.
Relationships among probability distributions
For more information, see normal approximation to binomial. For more information, see normal approximation to beta. For more information, see normal approximation to gamma. The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. The sum of the squares of n standard normal random variables is has a chi-squared distribution with n degrees of freedom.
For more information, see normal approximation to t. A chi-squared distribution with 2 degrees of freedom is an exponential distribution with mean 2. An exponential random variable with mean 2 is a chi-squared random variable with two degrees of freedom. More generally, sticking any random variable into its CDF yields a uniform random variable. More generally, applying the inverse CDF of any random variable X to a uniform random variable creates a variable with the same distribution as X.
The chi-square distribution with 2 degrees of freedom is the exponential distribution with scale parameter 2.
Diagram of distribution relationships
The chi-square distribution with 2 degrees of freedom is the gamma distribution with shape parameter 1 and scale parameter 2, which we already know is the exponential distribution with scale parameter 2.
Recall that if we add independent gamma variables with a common scale parameter, the resulting random variable also has a gamma distribution, with the common scale parameter and with shape parameter that is the sum of the shape parameters of the terms. Specializing to the chi-square distribution, we have the following important result: The last two results lead to the following theorem, which is fundamentally important in statistics.
This theorem is the reason that the chi-square distribution deserves a name of its own, and the reason that the degrees of freedom parameter is usually a positive integer.
The Chi-Square Distribution
Sums of squares of independent normal variables occur frequently in statistics. Here is the precise statement: Note the shape of the probability density function in light of the previous theorem. Like the gamma distribution, the chi-square distribution is infinitely divisible: Also like the gamma distribution, the chi-square distribution is a member of the general exponential family of distributions: