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Suppose that $X_1,X_2,\ldots$ are sequence of independent random variables with distribution Bernoulli($p$). if $ S_n=X_1+X_2+\ldots+X_k,N_k=\min\{n\ge 1,S_n=k \}.$ find distribution random varible $N_k.$

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    Tr$y$ readin$g$ about th$e$ [n$e$gative binomial distribution](http://en.wikipedia.org/wiki/Negative_binomial_distribution)2012-03-02

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Hint: The $k$-th success occurs on the $n$-th trial if and only if (i) there are exactly $k-1$ successes in the first $n-1$ trials and (ii) the $n$-th trial results in a success.

In the language of your question, we are tossing a fair die repeatedly. Let $X_i$ be $1$ if we toss a $6$ on the $i$-th trial, and let $X_i=0$ otherwise.

Let $S_n=X_1+X_2+\cdots +X_n$. Then $S_n$ counts the number of $6$'s in $n$ tosses.

Let $k$ be a fixed number, like $4$. Then $N_4$ is the smallest $n$ such that $S_n=4$. So it is the number of tosses we made until we got the fourth $6$. Calculate the probability that $N_4=17$. This is the probability that after $16$ tosses we were almost there (we had tossed exactly three $6$'s), and then we got a $6$ on the $17$-th toss.