I have a question regarding sum of Poisson random variables and using normal approximation: The number of whales spotted per day has Poisson Distribution $Po(1.2)$. The number of dolphins spotted per day has Poisson Distribution $Po(4.5)$. The two distributions are independent. For 100 consecutive days in a year, use approximation to find the probability that at least four times as many dolphins as whales are spotted in a year.
Let D represents the number of Dolphins, W represents the number of Whales. I know that sum of independent Poisson random variables is still Poisson. If suppose I am considering the random variable $X=4W-D$. (But I am not sure whether minus is allowed here) Then X follows Poisson Distribution $Po(30)$ which the $30$ is due to $100(1.2(4)-4.5)$. Then I am finding $P(X\leq0)$, which $X$ will have mean and variance $30$. Then I use normal approximation by doing the necessary continuity correction and so on.
However, if I do normal approximation directly first on the respective Poisson Distribution:
$Po(120)\sim N(120,120)$, $N$ is the normal distribution with mean $120$, variance $120$.
$Po(450)\sim N(450,450)$, $N$ is the normal distribution with mean $120$, variance $120$. Defining $X=4W-D$. Therefore $X$ follows $N(30,2370)$ and I am finding $P(X\leq0)$ using continuity correction and so on.
Both approaches yield very different answers.
Can someone rectify the error or explain the difference?