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Let $X_1,\dots,X_n$ are i.i.d random variables with geometric distribution, and the successful probability is $p$ for each $X_i$.

So for any $X_i$, the probability mass function is $\Pr(X_i=k)=(1-p)^{k-1}p$

Define a list of indicator variables $Y_1,\dots,Y_n$, which for an integer $b>0$, $$ Y_i=\left\{ \begin{aligned} 1 && X_i \le b\\ 0 && X_i >b \end{aligned} \right. $$

So my question is what is the distribution of $\sum\limits_{i=1}^n Y_i?$

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Each random variable $Y_i$ is a Bernoulli random variable with distribution $\mathrm P(Y_i=1)=r$ and $\mathrm P(Y_i=0)=1-r$ for a given parameter $r$ which you should be able to determine.

Hence, like any sum of $n$ i.i.d. Bernoulli random variables, $S_n=\sum\limits_{i=1}^nY_i$ is a binomial random variable with parameters $n$ and $r$. In other words, for every integer $k$ such that $0\le k\le n$, $$ \mathrm P(S_n=k)={n\choose k}r^k(1-r)^{n-k}. $$

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