Relationships Among Probability Distributions - Approximate (limit) Relationships

Approximate (limit) Relationships

Approximate or limit relationship means

• either that the combination of an infinite number of iid random variables tends to some distribution,
• or that the limit when a parameter tends to some value approaches to a different distribution.

Combination of iid random variables:

• Given certain conditions, the sum (hence the average) of a sufficiently large number of iid random variables, each with finite mean and variance, will be approximately normally distributed.(This is central limit theorem (CLT)).

Special case of distribution parametrization:

• X is a Hypergeometric (m, N, n) random variable. If n and m are large compared to N, and p = m / N is not close to 0 or 1, then X approximately has a Binomial(n, p) Distribution.

• X is a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large, then X approximately has a binomial(n, p) distribution.

• If X is a binomial (n, p) random variable and if n is large and np is small then X approximately has a Poisson(np) distribution.

• If X is a negative binomial random variable with r large, P near 1, and r(1-P) = λ, then X approximately has a Poisson distribution with mean λ.

Consequences of the CLT:

• If X is a Poisson random variable with large mean, then for integers j and k, P(j ≤ X ≤ k) approximately equals to P(j - 1/2 ≤ Y ≤ k + 1/2) where Y is a normal distribution with the same mean and variance as X.

• If X is a binomial(n, p) random variable with large n and np, then for integers j and k, P(j ≤ X ≤ k) approximately equals to P(j - 1/2 ≤ Y ≤ k + 1/2) where Y is a normal random variable with the same mean and variance as X, i. e. np and np(1-p).

• If X is a beta random variable with parameters α and β equal and large, then X approximately has a normal distribution with the same mean and variance, i. e. mean α/(α + β) and variance αβ/((α+β)2(α + β + 1)).

• If X is a gamma(α, β) random variable and the shape parameter α is large relative to the scale parameter β, then X approximately has a normal random variable with the same mean and variance.

• If X is a Student's t random variable with a large number of degrees of freedom ν then X approximately has a standard normal distribution.

• If X is an F(ν, ω) random variable with ω large, then ν X is approximately distributed As a chi-squared random variable with ν degrees of freedom.