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A note on negative λ-binomial distribution
Advances in Difference Equations volume 2020, Article number: 569 (2020)
Abstract
In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explicit expression for the moments of the negative λ-binomial random variable.
1 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
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
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
Recently, λ-analogue of binomial coefficients was considered by Kim to be
In this paper, we consider the negative λ-binomial distribution and obtain expressions for its moments.
2 Negative λ-binomial distribution
Definition 2.1
\(Y_{\lambda }\) is the negative λ-binomial random variable if the probability mass function of \(Y_{\lambda }\) with parameters \((r,p)\) is given by
where λ∈ (0,1) and p is the probability of success in the trials.
Note that
and
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 \(r(1-p)\).
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.
Theorem 2.1
Let \(Y_{\lambda }\) be a negative λ-binomial random variable with parameters \((r,p)\). Then we have
Note 2.1
where Y is the negative binomial random variable with parameters \((r,p)\).
Note 2.2
where Y is the Poisson random variable with parameter \(r(1-p)\).
Now, we observe that
The variance of random variable X is defined by
From Theorem 2.1, (11), and (12), we note that
Therefore, we obtain the following theorem.
Theorem 2.2
Let \(Y_{\lambda }\) be a negative λ-binomial random variable with parameters \((r,p)\). Then we have
Note 2.3
where Y is the negative binomial random variable with parameters \((r,p)\).
Note 2.4
where Y is the Poisson random variable with parameter \(r(1-p)\).
Note that
where \(S_{2}(n,l)\) is the Stirling number of the second kind, and
From (13), we note that
Therefore, we obtain the following theorem.
Theorem 2.3
Let \(Y_{\lambda }\) be a negative λ-binomial random variable with parameters \((r,p)\). Then we have
Note 2.5
where Y is the negative binomial random variable with parameters \((r,p)\) (see [4, 12]).
Note 2.6
where Y is the Poisson random variable with parameter \(r(1-p)\) (see [16]).
Note that
where \(Z_{\lambda }\) is the negative λ-binomial random variable with parameters \((r+\lambda ,p)\).
Therefore, we obtain the following theorem.
Theorem 2.4
Let \(Y_{\lambda }\), \(Z_{\lambda }\) be two negative λ-binomial random variables with parameters \((r,p)\), \((r+\lambda ,p)\) respectively. Then we have
3 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 λ-binomial random variable with parameter \((r,p)\) in (4) and deduced its expectation in Theorem 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.
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Acknowledgements
The authors thank Jangjeon Institute for Mathematical Science for the support of this research.
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This research was funded by the National Natural Science Foundation of China (No. 11871317, 11926325, 11926321).
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Ma, Y., Kim, T. A note on negative λ-binomial distribution. Adv Differ Equ 2020, 569 (2020). https://doi.org/10.1186/s13662-020-03030-z
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DOI: https://doi.org/10.1186/s13662-020-03030-z