3) A study is conducted to estimate the proportion of pine trees in a forest tha

Question

3) A study is conducted to estimate the proportion of pine trees in a forest that are infected by the pine shoot moth. The forest is divided into many, many 25’x25’ square sections. One hundred of these squares are selected at random, and the number of infected trees is recorded for each square.

(a) Let N = random number of trees in a random square. What is a good guess for the distribution of N?

(b) Let I = random number of infected trees in a random square. What is a good guess for the distribution of I?

(c) Let X = random number of the 100 squares which have at least one infected tree. What is a good guess for the distribution of X?

Explanation / Answer

a) A good guess for distribution N shall be 25 x 25 square section divided by average area taken up by a pine tree

b) A good guess can be taken by estimating standard probability of infected trees by refering past government data or surveys and using Binomial function wherin:-

N = Number of trees in 25 x 25 square section

P= Standard Probability of Infected Tree

(1-P)= Standard Probability of Healthy Tree/ Non Infected Tree.

C) Here we can increase the number of N by Multiplying 25 x 25 by 100

and again use binomial to make a guess of probability of having atleast one infected tree in 25x25 x100.

Since we have to take atleast one infected tree we can use negative binomial and taking all scenarios below 1 as a failure.

N = Number of trees in 25 x 25 square section x 100

P= Standard Probability of Infected Tree

(1-P)= Standard Probability of Healthy Tree/ Non Infected Tree.

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