Question

The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y^2 relates to the cost C completing this operation to the square of the time to completion. Find the mean and variance of C.

Answer 1

I have a mean of 1100. I need help with the Variance. I get so far and then I need and mgf, which i'm not sure how to calculate

Answer 2

@ganeshie8 - I was told maybe you can help me?

Answer 3

How did you get the mean?

Answer 4

E(100+40Y+3Y^2) = 100 + 40(E(Y)) +3(E(Y^2))

Answer 5

E(Y^2) = sd^2+mean^2, being exponential, mean is 10 and sd is 10^2 so E(Y^2) = 100+100 = 200. Insert into above equation and it comes to 1100

Answer 6

There are similar properties with variance as well.

Answer 7

The properties were unhelpful as far as I found anyways. :(

Answer 8

\[
\text{Var}(100 + 40Y + 3Y^2)=\text{Var}(100)+\text{Var}(40Y )+\text{Var}(3Y^2)
\]

Answer 9

This works, right?

Answer 10

no.

Answer 11

V(40Y+3Y^2) is what it would turn into. the Variance properties are different.

Answer 12

Okay, let \(X=Y^2\)\[
\text{Var}(40Y+3X) =40^2 \text{Var}(Y) +3^2\text{Var}(X) +2(40)(3)\text{Cov}(Y,X)
\]

Answer 13

I don't think that's very helpful either :/ I got it down to V(C) = E(C^2)-E(C)^2 by a property.

Answer 14

from my understanding I'd just need mgf's from there to calculate. My prof recommended this method.

Answer 15

You can do that, by there isn't anything tricky about my method.

Answer 16

I just don't know how to work your method either lol

Answer 17

The exponential distribution is: \[
\lambda e^{-\lambda x}
\]And the mean is \(1/\lambda\).

Answer 18

I'm not exactly certain how that helps me?

Answer 19

Because if you square it you can notice interesting things. \[
\lambda^2e^{-2\lambda x} = \frac \lambda 2(2\lambda e^{-2\lambda x)}
=\frac \lambda 2(\lambda' e^{-\lambda' x})
\]

Answer 20

I'm sorry (my brain has died) please explain this to me slowly and step by step. How does that help me? What do I do with it?