I have a question about my statistics homework.
The question is as follows:
At an army base there are X number of soldiers hit by a car. The poisson distribution expaction of this is μ=2. The number of soldiers Y get hit by a tank has a poisson distribution of λ=1. You can assume that no one will get hit by both(a car and tank). The random variables X and Y are independent.
(I) What is P(X+Y=k) for k=0,1,2,…?
Soldier who got hit by a tank have a probability of 1/2 to die. Z is the total amount of soldiers that die because of a hit by tank.
(II) What is P(Z=k) for k=0,1,2,…?
To Calculate the result I used the poisson formula: $$ e^{- \lambda} {\lambda^x\over x!} $$
In my solution I just used the X and Y, because I do not know if X or Y ends at 2 or continuous. I assumed that it continuous because at the end of the line you will see "...". This is my solution:
A) $$ P(x+y=k) = e^{- 2} {2^x\over x!} + e^{- 1} {1^y\over y!}$$ B) $$ P(z=k) = (e^{- 1} {1^z\over z!}) * 1/2 $$
I'm not sure if this is the correct answer. So I hope that someone could tell me if I did something wrong.
Hints:
(A) What is the expected number of vehicle hits per day? Does the number of vehicle hits per day have a Poisson distribution?
(B) What is the expected number of tank deaths per day? Does the number of tank deaths per day have a Poisson distribution?