Products on a production line are defective with probability $0.1$, stochastically independently of each other. Let $Y$ be the total number of products which an inspector checks who stops when the first defective product is found. Name the distribution which $Y$ follows. Give the corresponding weights, $E[Y]$ and $P(Y ≤ 2)$.
My answer:
Type of distribution: Y follows the geometric distribution with $Y $~$ geo(0.1)$.
Weights: For all natural numbers $k$, $P(Y=k) = (0.1)(0.9)^{k-1}$, and $0$ otherwise.
$E[Y] = \frac{1}{p} = 10 $
$P(Y \leq 2) = P(Y=1) + P(Y=2) = 0.1 + 0.09 = 0.19$
Are my answers correct ?