Common continuous distributions

Moving on to the common continuous families.

Uniform: f(x|a, b) = 1/(a - b)   x in [a, b]

Intuitive description: The analogous to the discrete uniform, the random variable is equally likely to be anywhere in the range.

Mean: (a + b)/2
Variance: (b - a)²/12

Gamma: f(x) = x^α-1e^-x / Γ(α), x>0  or, more commonly   f(x|α,β) = x^α-1e^-x/β / Γ(α)β^α

Intuitive description: The second parameterization is a bit easier to grasp (and, thus, more commonly used). The first parameter, α is a shape parameter which indicates the height of the peak. The second, β, determines the spread. Either way, it's a right-tailed distribution with a peak at (α-1)/β. The distribution is frequently used in modeling processes (known, shockingly, as gamma processes) where there is a varying degree of "memory" in the process. In the extreme case, the gamma yields the exponential, which is a memoryless distribution. The gamma also comes up in Bayesian stats a lot because it has nice conjugate prior properties (though that's less important now than it was 30 years ago, since current Bayesian work favors numerical, rather than analytic techniques.

Mean: α/β
Variance: α/β²

Chi-Squared: This is a gamma distribution with α = p/2 and β = 2. Here, p is a positive integer and is referred to as the "degrees of freedom".

Intuitive description: This represents the sum of squares from a sample of  p normal random variables. As such, it comes up a lot in regression analysis when assessing goodness of fit.

Mean: p
Variance: 2p

Exponential: This is a gamma distribution with α = 1. The spread is typically inverted to give a rate. Thus, f(x|λ) = λe^-λx.

Intuitive description: This is the memoryless version of the gamma. That is P(x>a|x>b) = P(x>a-b).

Mean: 1/λ
Variance: 1/λ²


Weibull: f(x|γ,β) = (γ/β)x^γ-1e^-xγ/β; x > 0, γ > 0, β > 0

Intuitive description: Can either be viewed as a general case of the exponential or a transformation of the exponential. If X ~ Exponential(λ) then X^1/γ ~ Weibull(γ,λ). Either way, it's a failure rate distribution.

Mean: γ Γ(1+1/β)
Variance: γ²[Γ(1+2/β) - Γ(1+1/β)²] (I have no intention of memorizing that)

Lots more to come, but that will do for today.