Question

1. In a southern pine stand infested by beetles, an average of 3 beetle larvae are found under a 20 cm by 10 cm square sample of bark. Let X = the number of beetle larvae found in a selected bark sample.
   1. What discrete probability distribution best describes X?  Name the distribution and write down the equation (with all parameters assigned).
   2. Find the probability that a randomly selected bark sample has:
      1. 1. Exactly 3 larvae
         1. 2. Less than 3 larvae
         2. 3. More than 3 larvae (hint: use the complement rule)

Answer 1

a)here since expected number of values are fixed and variable is independent from sample to sample .

X follows Poisson distribution with parameter =3

P(X=x) =e^-3*3^x/x!

b)

1) P( Exactly 3 larvae ) =P(X=3)=e^-3*3³/3! =0.2240

2) P( Less than 3 larva ) =P(X<3) =P(X=0)+P(X=1)+P(X=2)

=e^-3*3⁰/0! +e^-3*3¹/1! +e^-3*3²/2! =0.0498+0.1494+0.2240 =0.4232

c)

P( More than 3 larvae ) =P(X>=3)=1-P(X<=3)

=1-(e^-3*3⁰/0! +e^-3*3¹/1! +e^-3*3²/2!+e^-3*3³/3!)

=1-(0.0498+0.1494+0.2240+0.2240) =0.3528