Answer:
a) ![P(X=1) = \frac{e^{-10} 10^1}{1!}= 0.000454](https://tex.z-dn.net/?f=%20P%28X%3D1%29%20%3D%20%5Cfrac%7Be%5E%7B-10%7D%2010%5E1%7D%7B1%21%7D%3D%200.000454)
b) For this case the new parameter
would be:
![\lambda = 10 \frac{cars}{hour} * 3 hours =30](https://tex.z-dn.net/?f=%20%5Clambda%20%3D%2010%20%5Cfrac%7Bcars%7D%7Bhour%7D%20%2A%203%20hours%20%3D30)
And we want to calculate the following probability:
![P(X >29)](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29)
And we can use the complement rule for this case:
![P(X >29) =1-P(X\leq 29)](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29%20%3D1-P%28X%5Cleq%2029%29)
And for this case we can use the following excel code in order to find the required probability:
"=1-POISSON.DIST(29,30,TRUE)"
And we got: ![P(X >29) =1-P(X\leq 29)=0.5243](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29%20%3D1-P%28X%5Cleq%2029%29%3D0.5243)
Step-by-step explanation:
Part a
Let X the random variable that represent the number of cars arriving for gasoline at Shell station. We know that
The probability mass function for the random variable is given by:
And for this case we want this probability:
![P(X=1)](https://tex.z-dn.net/?f=%20P%28X%3D1%29)
Using the probability mass function we got:
![P(X=1) = \frac{e^{-10} 10^1}{1!}= 0.000454](https://tex.z-dn.net/?f=%20P%28X%3D1%29%20%3D%20%5Cfrac%7Be%5E%7B-10%7D%2010%5E1%7D%7B1%21%7D%3D%200.000454)
Part b
For this case the new parameter
would be:
![\lambda = 10 \frac{cars}{hour} * 3 hours =30](https://tex.z-dn.net/?f=%20%5Clambda%20%3D%2010%20%5Cfrac%7Bcars%7D%7Bhour%7D%20%2A%203%20hours%20%3D30)
And we want to calculate the following probability:
![P(X >29)](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29)
And we can use the complement rule for this case:
![P(X >29) =1-P(X\leq 29)](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29%20%3D1-P%28X%5Cleq%2029%29)
And for this case we can use the following excel code in order to find the required probability:
"=1-POISSON.DIST(29,30,TRUE)"
And we got: ![P(X >29) =1-P(X\leq 29)=0.5243](https://tex.z-dn.net/?f=%20P%28X%20%3E29%29%20%3D1-P%28X%5Cleq%2029%29%3D0.5243)