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If $X_1 ,X_2,X_3 \dots X_n $ are geometrically distributed iid's with parameter p. How do I calculate the probability density function of $ S_n = X_1+X_2 \dots + X_n$

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The probability that $S_n=m$ is the probability that the $n$-th success occurs on the $m$-th trial. This is the probability that there are exactly $n-1$ successes in the first $m-1$ trials, followed by a success on the $m$-th trial. Thus $$\Pr(S_n=m)=\binom{m-1}{n-1}p^{m-1}(1-p)^{m-n} p.$$ This expression can be simplified slightly.

It is required that $m\ge n$. Note that we have obtained the probability distribution function of $S_n$.

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