Gamma Distribution

**HoJo1001** · March 4th 2009, 06:29 PM

if we have these data points

4.413696 12.70539 14.2436 3.690484 14.02695 5.155644 1.657827 2.321249 5.034854 17.78997 14.17351 6.58265 14.53798 9.368805 13.46552 16.19866 8.830075 17.0933 5.909151 8.476882 16.87285 10.99451
and we assume its a gamma distribution, how can we calculate the 2 parameters of this distribution.

I'm not really sure how to figure this out so any suggestions would be appreciated.

**matheagle** · March 4th 2009, 10:45 PM

Originally Posted by HoJo1001

if we have these data points

4.413696 12.70539 14.2436 3.690484 14.02695 5.155644 1.657827 2.321249 5.034854 17.78997 14.17351 6.58265 14.53798 9.368805 13.46552 16.19866 8.830075 17.0933 5.909151 8.476882 16.87285 10.99451
and we assume its a gamma distribution, how can we calculate the 2 parameters of this distribution.

I'm not really sure how to figure this out so any suggestions would be appreciated.

I just covered method of moments today.
I would get the sample mean and sample second moment and solve for $\alpha$ and $\beta$ .
The population mean is $\alpha\beta$ .
And since the variance is $\alpha\beta^2$
you can compute $E(X^2)=V(X)+(EX)^2=\alpha\beta^2 +(\alpha\beta)^2$ .
I would set these equal to

${\sum_{k=1}^n X_k\over n}$ and ${\sum_{k=1}^n X^2_k\over n}$ .

But you can also go with MLEs too.

**HoJo1001** · March 5th 2009, 04:52 AM

So does that mean that
alpha = m1^2/(m2-m1^2)

and
Beta = (m2 - m1^2)/m1

where
m1 = x1 + x2 +...+ xn) /n

and

m2 = (x1^2 + x2^2 +.... xn^2) / n

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