# Probability distribution given a mean value

A certain section of a forest is undergoing a pine-beetle infestation. A biologist has determined that the number of pine-beetle infected trees fluctuates from acre to acre, with an average of $9.6$ pine-beetle infected trees per acre.

a) What is the probability that between $7$ and $9$, inclusive infected trees are found?

b) What is the probability that more than $12$ pine beetle infected trees are found?

c)As a way to combat the infestation, the infected trees are to be sprayed with an insecticide at a cost of $5$ $dollars$ for every tree infested with pine beetle(s), plus an overhead fixed cost of $70$ $dollars$ for equipment rental. Letting Cost represent the total cost for spraying all the pine-beetle infested trees for a randomly chosen acre of forest. Find the expected cost of spraying an acre. In addition, find the standard deviation in the cost of spraying an acre.

I'm pretty sure this is a normal distribution centered around the average value of 9.6, but using a normal distribution calculator, I'm not getting the correct answers.

• Have you learned about the Poisson Distribution? – Remy Mar 5 '18 at 5:17
• @Remy No! I guess this is what I use for this problem. I have a and b now, am just stuck on the standard deviation of the cost. – MattyS11 Mar 5 '18 at 5:27

Hint:

Let $K$ denote the number of infected trees found.

The Poisson Distribution has pmf

$$P(K=k)=\frac{\lambda^ke^{-\lambda}}{k!}$$

where

$$\lambda=9.6$$

For $(b)$ you'll want to note that

$$P(K>12)=1-P(K\leq12)$$

$(c)$

Let $C$ denote the total cost and let $T$ denote the total number of trees that need spraying. We have that

$$C=70+5T$$

Then by linearity of expectation

$$E(C)=E(70+5T)=70+E(5T)=70+5E(T)$$

By definition,

$$Var(\alpha X+\beta)=\alpha^2Var(X)$$

so we have that

$$Var(C)=Var(5T+70)=5^2Var(T)$$

From here, use the fact that the mean and variance are equal in the Poisson Distribution

• So, is the bottom equation the one for standard deviation? – MattyS11 Mar 5 '18 at 5:39
• That's the formula for the variance. Standard deviation is the square root of variance. – Remy Mar 5 '18 at 5:40
• How do i derive the standard deviation from the variance? – MattyS11 Mar 5 '18 at 5:42
• I just told you, no? – Remy Mar 5 '18 at 5:44
• Since the mean and the variance are equal, why isn't the answer just the square root of $5(E(T) + 70$? – MattyS11 Mar 5 '18 at 6:05