# X and Y are both geometric distributions with success p. What it the Pr(X+Y=n). Would I use a convolution with a sum for this? Do I need to define a third random variable?

Finding the probability distribution of the sum of geometric distributions?
X and Y are both geometric distributions with success p. What it the $Pr\left(X+Y=n\right)$
Would I use a convolution with a sum for this? Do I need to define a third random variable?
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asijikisi67
Alan Sherman
Step 1
From the definition of Geometric Random Variables:
If X,Y are the independent counts of trials, from sequences of Bernoulli trials, until first success (with rate p), then $X+Y$ would be the count of trials from a sequence of Bernoulli trials until the second success.
Step 2
We would then be seeking the probability of obtaining exactly one success somewhere among $n-1$ trials, then a second success.