2022-07-20

Let ${X}_{1},{X}_{2}$ be a random variables with geometric distribution with p,
Find the probability function of the random variable ${X}_{1}+{X}_{2}$

Expert

Step 1
The post contains some elements of a correct approach.
We cannot solve the problem without making some assumptions about the relationship between ${X}_{1}$ and ${X}_{2}$. We will assume that ${X}_{1}$ and ${X}_{2}$ are independent.
Let $S={X}_{1}+{X}_{2}$. We want to find $Pr\left(S=n\right)$. This is 0 if $n\le 1$. So let $n\ge 2$.
Step 2
We can have $S=n$ in several ways. For we could have ${X}_{1}=1$ and ${X}_{2}=n-1$. Or we could have ${X}_{1}=2$ and ${X}_{2}=n-2$. And so on up to ${X}_{1}=n-1$ and ${X}_{2}=1$.
For any k from 1 to $n-1$, we have $Pr\left({X}_{1}=k\cap {X}_{2}=n-k\right)=\left(1-p{\right)}^{k-1}p\left(1-p{\right)}^{n-k-1}p={p}^{2}\left(1-p{\right)}^{n-2}.$.
Adding up from $k=1$ to $k=n-1$ we find that $Pr\left(S=n\right)=\left(n-1\right){p}^{2}\left(1-p{\right)}^{n-2}.$

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