Answer:
(a) more than 2 such accidents in the next month ![\approx 0.3773](https://tex.z-dn.net/?f=%5Capprox%200.3773)
(b) more than 4 such accidents in the next 2 months ![\approx 0.44882](https://tex.z-dn.net/?f=%5Capprox%200.44882)
(c) more than 5 such accidents in the next 3 more than 5 such accidents in the next 3 months ![\approx 0.64533](https://tex.z-dn.net/?f=%5Capprox%200.64533)
Step-by-step explanation:
Let N be the Random variable that marks the number of crashes in certain month.
Now let us use Poisson distribution since we are given with average number of crashes that is N \sim Pois(2.2)
(A) more than 2 such accidents in the next month
Probability(more than 2 such accidents in the next month)=P(N>2)
P(N>2)=1-P(N=0)-P(N=1)-P(N=2)
=>![1-e^-{2.2}-2.2e^{-2.2}-\frac{2.2^2}{2!}e^{2.2}](https://tex.z-dn.net/?f=1-e%5E-%7B2.2%7D-2.2e%5E%7B-2.2%7D-%5Cfrac%7B2.2%5E2%7D%7B2%21%7De%5E%7B2.2%7D)
=> ![\approx 0.3773](https://tex.z-dn.net/?f=%5Capprox%200.3773)
B) more than 4 such accidents in the next 2 months
since the average number of crashes in 1 month is 2.2, the average number of crashes in two months is 4.4. hence, if we say that
is the number of crashes in 2 months, we have that
Pois(4.4)
Thus,
Probability(more than 4 such accidents in the next 2 months)=P(
)
=![1-P(N_1=0)-P(N_1=1)-P(N_1=2)=P(N_1=3)-P(N_1=4)](https://tex.z-dn.net/?f=1-P%28N_1%3D0%29-P%28N_1%3D1%29-P%28N_1%3D2%29%3DP%28N_1%3D3%29-P%28N_1%3D4%29)
![1-\sum_{k=1}^{4} \frac{4.4^{k}}{k !} e^{-4.4}](https://tex.z-dn.net/?f=1-%5Csum_%7Bk%3D1%7D%5E%7B4%7D%20%5Cfrac%7B4.4%5E%7Bk%7D%7D%7Bk%20%21%7D%20e%5E%7B-4.4%7D)
=> ![\approx 0.44882](https://tex.z-dn.net/?f=%5Capprox%200.44882)
C) more than 5 such accidents in the next 3 more than 5 such accidents in the next 3 months
If we say that
marks the number of crashes in the next 3 months , using the same argument as in (a) we have that a
Pois(6.6)
Hence
P(
)=![1-\sum_{k=0}^{5} \frac{6.6^{k}}{k !} e^{-6.6}](https://tex.z-dn.net/?f=1-%5Csum_%7Bk%3D0%7D%5E%7B5%7D%20%5Cfrac%7B6.6%5E%7Bk%7D%7D%7Bk%20%21%7D%20e%5E%7B-6.6%7D)
=>![\approx 0.64533](https://tex.z-dn.net/?f=%5Capprox%200.64533)