Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes

On Limited Length Binary Strings with an Application in Statistical Control

“…Proposition 1 (Makri and Psillakis [1]). For p (1) = = (p 0 , p 1 ) and the PMF f n;k,l (x), n ≥ k + 2 is given by:…”

“…, 0 ≤ k ≤ l ≤ n -2. The 0 -1 sequence n ≥ 2 on which M n;k,l is defined, is generated by a time-homogeneous first order Markov chain (MRKV) with one step transition probability matrix P = (p i,j ) and initial probability vector p (1) = with t ≥ 2, = P(X 1 = 0) = 1 -P(X 1 = 1) = 1 -and . Readily, a 0 -1 sequence , n ≥ 2 of independent and identically distributed (IID) RVs with a common probability of 1s, p = P(X i = 1) = 1 -P(X i = 0) = 1 -q, i = 1,2,...,n is a particular MRKV sequence with p 00 = p 10 = q, p 01 = p 11 = p and = (q, p).…”

“…Let A = {k + 2, k + 3,..., n} be an index set and {B, B c } a partition of A, where B = {k + 2, k + 3,..., l + 1} and B c = {l + 2, l + 3,..., n}. Then formally, M n;k,l , 0 ≤ k ≤ l ≤ n -2, can be written, on a 0 -1 sequence , as (see Makri and Psillakis [1]):…”

“…In Section 2 of the present paper we first introduce the required notation and second we recall a recent result of Makri and Psillakis [1] which refers to the probability mass function (PMF) of M n;k,l defined on a 0 -1 Markov chain. The presented expressions for the PMF are then used in an application case study which is discussed in Section 3 and it refers to statistical process control.…”

On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

“…Recently, Makri and Psillakis [28] obtained the probability mass function ; ,ℓ ( ) = ( ; ,ℓ = ), ∈ R( ; ,ℓ ) as well as the expected value ; ,ℓ = ( ; ,ℓ ) of ; ,ℓ defined on any 0-1 INID or EXCH sequence. Accordingly, with given by (5) and 6 (2) ,…”