I am solving past papers of my probability course. I don't have answer key to them. Here is the question I came across along with my approach. I think my approach is okay, but I am not confident. Kindly guide me if there is some correction needed. Thanks.
STATEMENT: An automated radar gun is placed on a road to record the speed of the cars passing by. The automated radar gun records $0.41$% of the cars going more than $20$ miles per hour above the speed limit. Assume the number of cars going more than $20$ miles above the speed limit has a Poisson distribution.
QUESTION: Find the value of paramter $\lambda$, mean & variance. In the last, calculate the probability that for $300$ randomnly chosen cars, more than $5$ of these cars will be exceeding the speed limit by more than $20$ miles per hour.
MY WORKING:
Let the $p$ denote the success rate of automated radar gun recording speed of the cars, then we are given that $p=0.41$, since the sample size $n=300$ is the big number and $p$ is smaller we can use poisson distribution to solve the problem (which we are already told to do in the statement part). Let $X$ denote poisson random variable then:
So in the case of approximating binomial distribution through poisson, we know that: $\lambda=np=300\times0.41=123$
Since mean & variance of poisson distribution are same, namely $\lambda$ we have:
$\lambda=123$,
mean=$\lambda=123$,
variance=$\lambda=123$
As for the last part, the probability that more than $5$ cars will be exceeding limit can be found as:
$P(X>5)=1-P(X\leq4)$ (which can be found by using poisson distribution with $\lambda=123$)
The question is very poorly phrased, as it could be interpreted 2 ways:
I will assume the latter of these as it makes more sense. Your approach is correct although notice that: $$0.41\%=\frac{0.41}{100}=0.0041$$ and so we should get: $$\lambda=np=300\times 0.0041=1.23$$ which makes a lot more sense as we know $\lambda<20$. Through the fact that less than $100\%$ are more than 20 over the limit