Take the 2-minute tour ×  Let $Y$ be the number of heads in a an
$X$ toss sequence of flipping a coin with probability $p$ of
heads. Show that $Y \sim \mathrm{Pois}(p \lambda)$ if $X \sim
\mathrm{Pois}(\lambda)$.  share|improve this question This looks
a lot like homework, and if so, please add the homework tag.
Hint: Use the law of total probability to calculate $P\{Y = k\}$
from $P\{Y = k\mid X = n\}$ and the known values of $P\{X =
n\}$. –  Dilip Sarwate Feb 19 '13 at 13:54 Sure. Right, so
that's what I have been doing, but it's not obvious. –  darksky
Feb 19 '13 at 13:56 Further hint: Can you tell what is the value
of $P\{Y=10\mid X = 4\}$? How about $P\{Y=10\mid X=15\}$? You
will need to use the fact that for a specific $k$, $P\{Y=k\mid
X=n\}$ is zero for some choices of $n$, cancel some factorial
terms in the rest, and sum a series. –  Dilip Sarwate Feb 19 '13
at 14:02 add comment  2 Answers  up vote 0 down vote accepted
Note that $Y\mid X=x\sim\mathrm{bi}(x,p)$ for $x=0,1,\ldots$ and
so $$ P(Y=k\mid X