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## Main Question or Discussion Point

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:

[tex]f\left(y;r,\lambda\right)=\frac{\lambda}{\Gamma\left(r\right)}\left(\lambda x\right)^{r-1}e^{-\lambda x}[/tex]

I also understand that the likelihood function is the product of the individual density functions.

Assuming independence, I write it as:

[tex]L\left(\underline{y};r, \lambda\right)=\left[f\left(y;r,\lambda\right)\right]^{n}[/tex]

[tex]=\left[\frac{\lambda^{r}y^{r-1}e^{-\lambda y}}{\Gamma\left(r\right)}\right]^{n}[/tex]

I am now stuck with the product of the [tex]y^{r-1}[/tex] and [tex]\Gamma\left(r\right)[/tex].

Please help me what to do, since I need the answer to find the maximum likelihood estimator of [tex]\lambda[/tex].

I understand that the density function is the following:

[tex]f\left(y;r,\lambda\right)=\frac{\lambda}{\Gamma\left(r\right)}\left(\lambda x\right)^{r-1}e^{-\lambda x}[/tex]

I also understand that the likelihood function is the product of the individual density functions.

Assuming independence, I write it as:

[tex]L\left(\underline{y};r, \lambda\right)=\left[f\left(y;r,\lambda\right)\right]^{n}[/tex]

[tex]=\left[\frac{\lambda^{r}y^{r-1}e^{-\lambda y}}{\Gamma\left(r\right)}\right]^{n}[/tex]

I am now stuck with the product of the [tex]y^{r-1}[/tex] and [tex]\Gamma\left(r\right)[/tex].

Please help me what to do, since I need the answer to find the maximum likelihood estimator of [tex]\lambda[/tex].