Discrete Random Variables

$P(X=3) = q^2 p = 0.8^2 \times 0.2 = 0.128$.

$E(X) = 1/p = 4$. $P(X > 4) = q^4 = 0.75^4 = 0.3164$ — the first four trials all failed.

$P(X \le 1) = e^{-4}(1 + 4) = 5e^{-4} = 0.0916$.

$X \sim \text{Po}(1.5)$, so $P(X=2) = \dfrac{e^{-1.5}(1.5)^2}{2!} = 0.2510$.

Independent Poissons add: the total is $\text{Po}(10)$. $P(X=8) = \dfrac{e^{-10}10^8}{8!} = 0.1126$.

$np = 2 < 5$ with $n$ large, so $X \approx \text{Po}(2)$ and $P(X=0) = e^{-2} = 0.1353$.