Answer:
a) 0.138
b) 0.938
c) 0.544
d) 6
e) 2.45
Step-by-step explanation:
Let X represent the number of tracks counted in 1 cm^2 of surface area.
The estimated average is 6 tracks per cm^2. This is the parameter λ of the Poisson distribution.
The expression for the Poisson distribution is
[tex]P(x=k)=\frac{\lambda^ke^{-\lambda}}{k!}[/tex]
Then
a) x=7
[tex]P(x=7)=\frac{\lambda^7e^{-\lambda}}{7!}=\frac{6^7e^{-6}}{7!}=\frac{693.89}{5040}=0.138[/tex]
b) x≥3
[tex]P(x\geq 3)=1-(P(0)+P(1)+P(2))\\\\P(0)=\frac{6^0e^{-6}}{0!}=\frac{0.00248}{1} =0.00248\\\\P(1)=\frac{6^1e^{-6}}{1!}=\frac{0.01487}{1} =0.01487\\\\P(2)=\frac{6^2e^{-6}}{2!}=\frac{0.0892}{2} =0.04462\\\\\\P(x\geq3)=1-(0.00248+0.01487+0.04462)=1-0.06197=0.938[/tex]
c) 2<x<7
[tex]P(2 < X < 7)= P(x=3)+P(x=4)+P(x=5)+P(x=6)\\\\P(x=3)=0.0892\\\\P(x=4)=0.1339\\\\P(x=5)=0.1606\\\\P(x=6)=0.1606\\\\\\P(2 < X < 7)=0.0892+0.1339+0.1606+0.1606=0.5443[/tex]
d) and e)
[tex]\mu=\lambda=6\\\\\\\sigma=\sqrt{\lambda}=2.45[/tex]
