- #1
Ad VanderVen
- 169
- 13
- TL;DR Summary
- I am trying to derive the convolution of two geometric distributions but obviously I am making an error.
I'm trying to derive the convolution from two geometric distributions, each of the form:
$$\displaystyle \left( 1-p \right) ^{k-1}p$$ as follows $$\displaystyle \sum _{k=1}^{z} \left( 1-p \right) ^{k-1}{p}^{2} \left( 1-p \right) ^{z-k-1}.$$ with as a result: $$\displaystyle \left( 1-p \right) ^{z-2}{p}^{2}z$$ Now the sum: $$\displaystyle \sum _{z=2}^{\infty } \left( 1-p \right) ^{-2+z}{p}^{2}z.$$ should be equal to ##\displaystyle \frac{2}{p}## which is not the case.
What am I did wrong?
$$\displaystyle \left( 1-p \right) ^{k-1}p$$ as follows $$\displaystyle \sum _{k=1}^{z} \left( 1-p \right) ^{k-1}{p}^{2} \left( 1-p \right) ^{z-k-1}.$$ with as a result: $$\displaystyle \left( 1-p \right) ^{z-2}{p}^{2}z$$ Now the sum: $$\displaystyle \sum _{z=2}^{\infty } \left( 1-p \right) ^{-2+z}{p}^{2}z.$$ should be equal to ##\displaystyle \frac{2}{p}## which is not the case.
What am I did wrong?
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