Question

the number of parasites on fish hatched inthe same season and living in the same farm pond follows a poisson distribution with a mean of 3.8 parasites per fish. What is the probability : a) a randomly selected fish will have more than 4 parasites? b) exactly 2 parasites? c) what is the expected number of parasites on a randomly selectedfish?

Answer 1

Let \(X\) denote the number of parasites found on a fish. Given that \(X\) follows a Poisson distribution with mean \(\lambda=3.8\), you immediately know that \(E(X)=\lambda=3.8\), which answers part (c).
\[(a)~~P(X>4)\\
\begin{align*}P(X>4)&=1-P(X\le4)\\
&=1-\sum_{x=0}^4\frac{e^{\lambda}\lambda^x}{x!}\\
&=1-\frac{e^{3.8}3.8^0}{0!}-\frac{e^{3.8}3.8^1}{1!}-\frac{e^{3.8}3.8^2}{2!}-\frac{e^{3.8}3.8^3}{3!}-\frac{e^{3.8}3.8^4}{4!}\\
&=\cdots
\end{align*}\]
\[(b)~~P(X=2)\\
P(X=2)=\frac{e^{\lambda}\lambda^2}{2!}=\cdots\]

Answer 2

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