tailieunhanh - Moments equalities for nonnegative integer-valued random variables

We present and prove two theorems about equalities for the nth moment of nonnegative integer-valued random variables. These equalities generalize the well known equality for the first moment of a nonnegative integer-valued random variable X in terms of its cumulative distribution function, or in terms of its tail distribution. | Turk J Math 28 (2004) , 111 – 117. ¨ ITAK ˙ c TUB Moments Equalities for Nonnegative Integer-Valued Random Variables Mohamed I. Riffi Abstract We present and prove two theorems about equalities for the nth moment of nonnegative integer-valued random variables. These equalities generalize the well known equality for the first moment of a nonnegative integer-valued random variable X in terms of its cumulative distribution function, or in terms of its tail distribution. Key words and phrases: Expectation, Moments, Equalities. 1. Introduction There is a well-known equality for the nth moment of a nonnegative random variable Y as an integral of a function of its tail distribution. A similar equality for the first moment of a nonnegative integer-valued random variable as a sum over x of a function of its tail distribution is also well known and used a lot in the literature (See [1, p. 43], for example). What we prove in this paper is a generalization of this sum equality when the random variable is integer-valued. In the next section we will prove a generalization of the well-known equality in the discrete case. Our equality gives a neat formula of the nth moment, when it exists, for nonnegative integer-valued variables. 2000 Mathematics Subject Classification: Primary 60A99. Secondary 62B99. 111 RIFFI 2. Main Theorems In this section we prove two identities that each will be used to prove one of our main theorems. The first identity is used to express a product of terms of the form (X − i) as a finite sum of products of similar terms when the sum ranges from 1 to a nonnegative integer x. The second identity is used to express xn as a finite sum that ranges from 1 to a nonnegative integer x. Before we proceed to the main theorems, we need the following lemma. Lemma Let x and n be nonnegative integers such that x > n. Then x n+1 Proof. = x−n X j=1 x−j . n Apply Pascal’s identity, namely x x−1 x−1 = + , n+1 n n+1 to each last .