The resulting pmf is given by Pleft(Y=kright)=frac{1}{ log;left(p qright)};left{begin{array}{l} log pleft[{q}^{-left k+1right)}left(1-qright)right],;k=cdots, -2,-1 log;qleft[{p}^{-left k+1rightright],kern1.08em k=0,1,2,cdots end{array}left 1-qright
Let X be a Gaussian random variable with expectation μ and variance σ 2. Define the quantized Gaussian random variable as follows: Y=⌊X/Δ⌋; its pmf is given by (P_{mu,sigma } = {p_{mu,sigma }[!k]}_{k=-infty }^{infty }) with: begin{aligned} p_{mu,sigma}[!k] &= mathbb{P}[!Y=k] &= int_{Delta k-1/2)}^{Delta k-1/2)} frac{1}{sigmasqrt{2pi}} expleft(-frac{(x-mu)^{2}}{2sigma^{2}} right) Delta k+1/2
The cumulative distribution function (cdf) of KGD and its corresponding probability mass function (pmf) are given, respectively, by Gleft xright) =1-left[ 1-left 1-q^{x+1-left 1-q^{pha }right]^{beta },,x=0,1-left 1-q^{).
Based on the same lines as the derivation in (4), their PMFs are given in the following lemma.
Their PMFs are given by (12) and (13).
(14) The pmf on c is given by the recursion (15) with P c(1 | g,1) = 1 and P c(c | g,1) = 0 for c ≠ 1. P S(S | c) is approximated by minimizing the number, m, of mutations over the assignments of observed sequences to c groups.
When applying early stop, the pmf of message complexity is given by replacing the upper limit s in (4) with z : = min ( { 1 ≤ t ≤ s | x t = 1 } ∪ { s } ).
If the underlying continuous random variable X has the cdf F X (x) = Pr(X ≤ x) then the pmf of the discrete analogue Y is given by Pleft(Y=kright)={F}_Xleft k+delta right -{F}_Xleft k-left[1-delta right -{F}_Xleft k-left[1-delta
The pmf of the discrete power distribution is given by Pleft(Y=kright)=left{begin{array}{l}frac{{left k-a+1right)}{l}frac{{left k-a+1righteft(b-aright);{left(m-aright)}^n-{left k-aright,k=a,;a+1, dots,;m-1 frac{{left(b-kright)}^n-{left k-arightht)}^n-{left k-aright);{left(b-mright)}^nn-1}{left b-arightk=m,;m+1, dots, b-1end{array}{left b-aright
The position of the minima arising from these simulations versus the classical PMF simulations described above is given in Figure 6 of the Supporting Information.
