More continuous distributions

Beta: f(x|α,β) = x^α-1(1-x)^β-1 / B(α,β), 0 <  x < 1, α > 0, β > 0

Intuitive description: This comes up a lot in Bayesian modeling, since it's a very convenient prior for a parameter that has a finite interval. The parameters give tremendous flexibility over the shape and it's a fairly easy distribution to compute. In many cases, it gives a conjugate prior (that is, the posterior distribution is of the same form).

Mean: α / (α + β)
Variance: αβ /[(α + β)²(α + β + 1)]

Double Exponential: f(x|μ,σ) = (1/2σ)e^-|x-μ|/σ

Intuitive description: Here, the absolute difference between the random variable and it's mean is exponentially distributed. The resulting distribution is symmetric, but with "heavy tails", that is, the distribution has less mass concentrated around the mean than the normal distribution. Unlike some heavy tailed, distributions, the double exponential does have all it's moments.

Mean: μ
Variance: 2σ²

Cauchy: f(x|θ) = 1/[π(1 + (x - theta)²)]

Intuitive description: The king of the heavies. This distribution is spread so wide it has no moments, not even a mean. While largely theoretical (the lack of moments makes it useful as an edge case when testing ideas), it does come up in real life: the quotient of two standard normals is Cauchy.

Mean: none!
Variance: none!