A sequence of independent Bernoulli trials, each of which is a success with probability $p$, is conducted. For $k\in {\mathbb{Z}}^{+}$, let $X_k$ 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 $X_k$, 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 $n$th factorial moment of $X_k$.

中文翻译：

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关于需要获得的试验次数 $ķ$ 连续成功

一系列独立的伯努利试验，每个试验都有可能成功 $p$进行。为了$ķ\in {\mathbb{ž}}^{+}$， 让 $X_{ķ}$ 是获得所需的试验次数 $ķ$连续的成功。使用基本概率论的技术，我们给出了一个推导，最终得出了概率概率函数的优雅表达。$X_{ķ}$，并且与文献中的内容相比更简单。此后，我们使用派生公式为互补累积分布函数和$ñ$的阶乘矩 $X_{ķ}$。