# Thread: Exponential dsn - expected cost of a project

1. ## Exponential dsn - expected cost of a project

Hi all,

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

So mean of C;
$\displaystyle E(C)=E(100+40Y+3Y^2)=100+40\cdot E(Y)+3\cdot [E(Y)]^2=100+40\cdot 10+3\cdot 100=800$

Just not sure I have calculated variance correctly;$\displaystyle E(C^2)=E[(100+40Y+3Y^2)^2]=$$\displaystyle E(100^2+1600Y^2+9Y^4+8000Y+600Y^2+240Y^3)=640000 So \displaystyle var(C)=640000-800^2=0 2. Originally Posted by Robb Hi all, The length of time \displaystyle Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula \displaystyle C=100+40Y+3Y^2 relates the cost \displaystyle C of completeting this operation to the square of the time to completion. Find the mean and variance of \displaystyle C So mean of C; \displaystyle E(C)=E(100+40Y+3Y^2)=100+40\cdot E(Y)+3\cdot [E(Y)]^2=100+40\cdot 10+3\cdot 100=800 Just not sure I have calculated variance correctly;\displaystyle E(C^2)=E[(100+40Y+3Y^2)^2]=$$\displaystyle E(100^2+1600Y^2+9Y^4+8000Y+600Y^2+240Y^3)=640000$

So $\displaystyle var(C)=640000-800^2=0$
You have made the very bad mistake of thinking that $\displaystyle E(Y^n) = [E(Y)]^n$.

You're given $\displaystyle E(Y) = 10$. To get $\displaystyle E(Y^n)$ you need to use the pdf: $\displaystyle E(Y^n) = \int_0^{+\infty} y^n f(y) \, dy$.

$\displaystyle E(C) = E(100 + 40Y + 3Y^2) = 100 + 40 E(Y) + 3 E(Y^2)$.

$\displaystyle Var(C) = E(C^2) - [E(C)]^2$

and note that $\displaystyle C^2 = (100 + 40Y + 3Y^2)^2$.