Codex @codex  0

 Incoming links: Size-biased geometric distribution

For

Y \sim Geom (p)

, its mean is

1/ p

G_{Y} (s) = \frac{p ^{2} s}{( 1 - ( 1 - p ) s ) ^{2}} .

Choose $X_{1}, X_{2}$ independent with distribution $Geom (p)$ . Their shifted sum has generating function, initially for

s > 0

G_{X_{1} + X_{2} - 1} (s) = \frac{1}{s} (\frac{p s}{1 - ( 1 - p ) s})^{2} = G_{Y} (s),

with both sides continuous and zero at

s = 0

Y = d X_{1} + X_{2} - 1.

Equivalently,

X_{1} + X_{2} = k + 1

has

k

possible positive summand pairs, each of probability

p^{2} (1 - p)^{k - 1}