Question

1. Assume that the number of episodes per year of otitis media, a common disease of the middle ear in early childhood follows a Poisson distribution with 2.7 episodes per year. What is the probability of having at most 2 episodes in the first year of life? (Approximate your answer to 4 decimal places).
2. Assume that the number of episodes per year of otitis media, a common disease of the middle ear in early childhood follows a Poisson distribution with 1.35 episodes per year. What is the probability of having 3 episodes in the first two years of life?  (Approximate your answer to 4 decimal places).

Answer 1

$1:\\X\sim Poiss\left( 2.7 \right) \\P\left( X\leqslant 2 \right) =P\left( X=0 \right) +P\left( X=1 \right) +P\left( X=2 \right) =\\=\sum_{i=0}^2{e^{-2.7}\frac{2.7^i}{i!}}=e^{-2.7}\left( 1+2.7+\frac{2.7^2}{2} \right) =0.4936\\2:\\X_2\sim Poiss\left( 1.35\cdot 2 \right) =Poiss\left( 2.7 \right) \\P\left( X_2=3 \right) =e^{-2.7}\frac{2.7^3}{3!}=0.2205$