safina
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There is a random sample of size n from a gamma distribution, with known r. Please help me formulate the likelihood function of the gamma distribution.
I understand that the density function is the following:
f\left(y;r,\lambda\right)=\frac{\lambda}{\Gamma\left(r\right)}\left(\lambda x\right)^{r-1}e^{-\lambda x}
I also understand that the likelihood function is the product of the individual density functions.
Assuming independence, I write it as:
L\left(\underline{y};r, \lambda\right)=\left[f\left(y;r,\lambda\right)\right]^{n}
=\left[\frac{\lambda^{r}y^{r-1}e^{-\lambda y}}{\Gamma\left(r\right)}\right]^{n}
I am now stuck with the product of the y^{r-1} and \Gamma\left(r\right).
Please help me what to do, since I need the answer to find the maximum likelihood estimator of \lambda.
I understand that the density function is the following:
f\left(y;r,\lambda\right)=\frac{\lambda}{\Gamma\left(r\right)}\left(\lambda x\right)^{r-1}e^{-\lambda x}
I also understand that the likelihood function is the product of the individual density functions.
Assuming independence, I write it as:
L\left(\underline{y};r, \lambda\right)=\left[f\left(y;r,\lambda\right)\right]^{n}
=\left[\frac{\lambda^{r}y^{r-1}e^{-\lambda y}}{\Gamma\left(r\right)}\right]^{n}
I am now stuck with the product of the y^{r-1} and \Gamma\left(r\right).
Please help me what to do, since I need the answer to find the maximum likelihood estimator of \lambda.