Answer:
P(x < 3) = 0.42319
Step-by-step explanation:
From the given information:
The mean density [tex]\lambda =[/tex] 3
Let x be the random variable that follows a Poisson distribution.
Therefore:
[tex]\mathtt{P(x) = \dfrac{e^{-\lambda} \lambda ^x}{x!}}[/tex] for x =1, 2, 3...
However, the probability that a random quadrat contains less than 3 spatuletails can be computed as:
[tex]\mathtt{P(x <3 ) = \dfrac{e^{-3}3 ^0}{0!}+ \dfrac{e^{-3}3 ^1}{1!}+ \dfrac{e^{-3} 3 ^2}{2!} }[/tex]
[tex]\mathtt{P(x <3 ) =e^{-3} \begin{pmatrix} \dfrac{3 ^0}{0!}+ \dfrac{3 ^1}{1!}+ \dfrac{3 ^2}{2!} \end {pmatrix} }[/tex]
[tex]\mathtt{P(x <3 ) =e^{-3} \begin{pmatrix} \dfrac{1}{1}+ \dfrac{3 }{1}+ \dfrac{9}{2} \end {pmatrix} }[/tex]
[tex]\mathtt{P(x <3 ) =e^{-3} \begin{pmatrix} 1+3+ 4.5 \end {pmatrix} }[/tex]
[tex]\mathtt{P(x <3 ) =e^{-3} \begin{pmatrix}8.5 \end {pmatrix} }[/tex]
P(x < 3) = 0.42319
