Variance by strata

It's just symbol manipulation, but there's enough of it to be worth writing down. So, I'll write it here. This is the derivation of the variance for an individual observation from stratum k.

P(X = 2^k) = θλ
P(X = -2^k) = θ(1-λ)
P(X = 0) = 1-θ

stratum mean μ = ΣxP(x) = 2^kθλ - 2^kθ(1-λ) = 2^kθ(2λ - 1)

stratum variance σ² = Σ(μ-x)²P(x)

   = θλ(μ - 2^k)² + θ(1 - λ)(μ + 2^k)² + (1 - θ)μ²

   = θλμ² - 2θλμ2^k + θλ2^2k + θ(1-λ)μ² + 2θ(1-λ)μ2^k + θ(1-λ)2^2k + (1-θ)μ²

   = μ² + 2θ(1-2λ)μ2^k + θ2^2k

Substituting in the formula for the mean to get the variance just in terms of θ and λ is not enlightening and doesn't help the computational stability, so I'll leave it as an exercise to any reader that just really likes moving letters around.