Multimodality of the Markov Binomial Distribution

Abstract

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closedform expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.

A Simple Stochastic Kinetic Transport Model

“…We then recover a (more general and more detailed) version of the main result of Gut and Ahlberg [6, p. 251]. In a similar way as for Proposition 7.1, the mean and variance of S F n (t) can be obtained by using (5) and Proposition 3.1 of [4] together with (3.1). Substituting (5.1) into the moments of S F n (t) and letting n → ∞, we obtain the following result.…”

“…The distribution of K n is then well known, and is called a Markov binomial distribution (MBD) (see, e.g. [4] and [10]). Clearly, the stationary distribution (π F , π A ) of the Markov chain {Y k , k ≥ 1} is given by π F = b/(a + b) and π A = a/(a + b).…”

“…We can then write P ν (Y k = τ ) = π τ (1 − ε τ γ k−1 ) for k ≥ 1, where γ = 1 − a − b is the smallest eigenvalue of P (see also [4] for the computations).…”

“…[2,Theorem 25.12]) by the easily proved uniform integrability of S n (t), (S n (t)) 2 , S τ n (t), and (S τ n (t)) 2 , which is implied by the uniform boundedness of (S n (t)) 3 and (S τ n (t)) 3 . Since X k is independent of K n , from (4) and Proposition 2.1 of [4] together with (3.1), we can determine the first and second moments of S n (t). The following proposition can be obtained by substituting (5.1) into the moments of S n (t) and letting n → ∞.…”

“…is a mixture of Gaussian distributions with mean jv t and variance 2jD t. Recall from [4] that the probability mass function f F n of K F n can be unimodal or bimodal. This property of K F n gives rise to the same phenomenon for S F n (t), i.e.…”