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On the number of trials needed to obtain k consecutive successes

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  • Drekic, Steve
  • Spivey, Michael Z.

Abstract

A sequence of independent Bernoulli trials, each of which is a success with probability p, is conducted. For k∈Z+, let Xk be the number of trials required to obtain k consecutive successes. Using techniques from elementary probability theory, we present a derivation which ultimately yields an elegant expression for the probability mass function of Xk, and is simpler in comparison to what is found in the literature. Following this, we use our derived formula to obtain explicit closed-form expressions for the complementary cumulative distribution function and the nth factorial moment of Xk.

Suggested Citation

  • Drekic, Steve & Spivey, Michael Z., 2021. "On the number of trials needed to obtain k consecutive successes," Statistics & Probability Letters, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:stapro:v:176:y:2021:i:c:s0167715221000948
    DOI: 10.1016/j.spl.2021.109132
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    References listed on IDEAS

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    1. Philippou, Andreas N. & Georghiou, Costas & Philippou, George N., 1983. "A generalized geometric distribution and some of its properties," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 171-175, June.
    2. Mitchell, Douglas W., 2000. "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control, Elsevier, vol. 24(4), pages 615-622, April.
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