# Variance of a poisson distribution

For a certain section of pine forest, the number Y of diseased trees per acre has a Poisson distribution with mean lambda=10. The disased trees are sprayed with an insecticide at a cost of 3 dollars per tree, plus a fixed overhead cost for equipment rental of 50 dollars. Letting C denote the total spraying cost for a randomly selected acre, find the expected value and variance for C.

Expected value is easy to find: 80

I'm stuck on Variance? Why is variance 90?

Because $C = 3Y+50$, and $var(C)=var(3Y+50)=9var(Y)=90$

• Thank you so much. Just a doubt, if we had to do it by the method of EX^2 etc. how would it look like?
– user218970
Sep 9, 2018 at 17:31
• $E[C^2]=E[(3Y+50)^2]=E[9Y^2+300Y+2500]=9E[Y^2]+300E[Y]+2500$; since Possion variance is $\lambda = 10$, $E[Y^2] = var(Y) + E[Y]^2 = 110$. Substituting leaves us with $E[C^2] = 6490$. $E[C]^2 = 6400$, then $var(C) = 6490-6400=90$ Sep 9, 2018 at 17:36
• Wow, thank you so much. I was stuck with this for minutes.
– user218970
Sep 9, 2018 at 17:40
• I did upvote it. But it won't show as I have less than 50 reputation. Yes, approved it.
– user218970
Sep 9, 2018 at 17:54
• You need a mere 15 reputation to upvote, not 50. [A single question, moderately well-written -- i.e. a somewhat average question on the site - can expect to earn about 2-3 upvotes, and 3 is sufficient to gain the privilege] Sep 10, 2018 at 1:03