# gamma dist

• Oct 27th 2010, 11:22 AM
sfspitfire23
gamma dist
Hey guys-been struggling with this question the whole day-

Weekly downtime X (in hrs) for a certain industrial machine has approximately a gamma dist. with alpha=3.5 and beta=1.5. Loss L (in dollars) as a result of this downtime is given by $\displaystyle L=30X+2x^2$.

What is the expected value and variance of L?

So, I first set out to find an equation for $\displaystyle E(X^k)$ and I ended up with $\displaystyle \frac{\beta^k \Gamma(\alpha+k)}{\Gamma(\alpha)}$ and I am unsure how to proceed from here. Also, is my derivation correct?
• Oct 27th 2010, 11:27 AM
mr fantastic
Quote:

Originally Posted by sfspitfire23
Hey guys-been struggling with this question the whole day-

Weekly downtime X (in hrs) for a certain industrial machine has approximately a gamma dist. with alpha=3.5 and beta=1.5. Loss L (in dollars) as a result of this downtime is given by $\displaystyle L=30X+2x^2$.

What is the expected value and variance of L?

So, I first set out to find an equation for $\displaystyle E(X^k)$ and I ended up with $\displaystyle \frac{\beta^k \Gamma(\alpha+k)}{\Gamma(\alpha)}$ and I am unsure how to proceed from here. Also, is my derivation correct?

$\displaystyle \displaystyle E(L) = 30 E(X) + 2 E(X^2)$. I haven't time to check your expression for E(X^k) but I'm sure there are websites that would either confirm or deny the result.
• Oct 27th 2010, 09:50 PM
matheagle
In order to obtain the variance of L, first obtain it's second moment.

$\displaystyle E(L^2)=900E(X^2)+120E(X^3)+4E(X^4)$

Remember that the beta can either be in the numerator or denominator, depending on how you write your density.
The technique is easy. OR just the MGF and differentiate.