Question

(34) Continued... Suppose and have geometric distributions with different parameters Geometric($\alpha$) and Geometric($\beta$). Find the convolution of their marginal distribution^1.

1.

n-11-a

Answer 1

(by law of total probability)

2-1 r=1

$= \sum_{x=1}^{n-1}P(x+Y=n)P(X=x)$

2-1 r=1

(since distribution of X and Y are geometric, given)

2-1 r=1

$= \alpha \beta (1-\beta )^{n-2}\sum_{x=1}^{n-1}\frac{(1-\alpha)^{x-1}}{(1-\beta )^{x-1}}$

72-1 1-B 1-5

α-β

where , n=2,3,.......