A note on negative λ-binomial distribution

and we say that X has a negative binomial distribution with parameters (r, p) (see [1– 3, 12, 13]). The negative binomial distribution is sometimes defined in terms of the random variable Y , the number of failures before the rth success. This formulation is statistically equivalent to one given above in terms of X denoting the trial at which the rth success occurs, since Y = X – r. The alternative form of the negative binomial distribution is


Introduction
In a sequence of independent Bernoulli trials, let the random variable X denote the trial at which the rth success occurs, where r is a fixed nonnegative integer. Then x -1 r -1 p r (1p) x-r , x = r, r + 1, r + 2, . . . , and we say that X has a negative binomial distribution with parameters (r, p) (see [1-3, 12, 13]). The negative binomial distribution is sometimes defined in terms of the random variable Y , the number of failures before the rth success. This formulation is statistically equivalent to one given above in terms of X denoting the trial at which the rth success occurs, since Y = Xr. The alternative form of the negative binomial distribution is where p is the probability of success in the trial (see [1,3,12,13]). It is known that the degenerate exponential function is defined by where (x) 0,λ = 1, (x) n,λ = x(xλ) · · · x -(n -1)λ (n ≥ 1) (see [5-7, 10, 11]).
Recently, λ-analogue of binomial coefficients was considered by Kim to be [6,8,9]). (3) In this paper, we consider the negative λ-binomial distribution and obtain expressions for its moments.
2 Negative λ-binomial distribution Definition 2.1 Y λ is the negative λ-binomial random variable if the probability mass function of Y λ with parameters (r, p) is given by where λ ∈ (0,1) and p is the probability of success in the trials.
From (4), we note that is the probability mass function of negative binomial random variable with parameters (r, p), and is the probability mass function of Poisson random variable with parameters r(1p). Let X be a discrete random variable, and let f (x) be a real-valued function. Then we have where p(x) is the probability mass function. From (9), we note that . Therefore, by (10), we obtain the following theorem. .
where Y is the negative binomial random variable with parameters (r, p).
where Y is the Poisson random variable with parameter r(1p).
Now, we observe that .
The variance of random variable X is defined by (see [1,3]).

Theorem 2.2 Let Y λ be a negative λ-binomial random variable with parameters (r, p). Then we have
where Y is the negative binomial random variable with parameters (r, p).
where Y is the Poisson random variable with parameter r (1p).

Theorem 2.3
Let Y λ be a negative λ-binomial random variable with parameters (r, p).
where Y is the negative binomial random variable with parameters (r, p) (see [4,12]).
where Y is the Poisson random variable with parameter r(1p) (see [16]).

Conclusion
In this paper, we introduced one discrete random variable, namely the negative λ-binomial random variable. The details and results are as follows. We defined the negative λbinomial random variable with parameter (r, p) in (4) and deduced its expectation in The-orem 2.1. We also obtained its variance in Theorem 2.2 and derived explicit expression for the moment of the negative λ-binomial random variable in Theorem 2.3.