How to find the mode and median of a Gamma distribution? A random variable has Gamma distribution with mean of $10$ and standard deviation of $5$. The mode and median are to be found. 
I realize that this means that $\alpha$ and $\beta$ are both $\sqrt{5}$. My professor told me that R is needed for one of them, and the exact answer can be found another way. Please help. I am confused at what to do.
 A: EDIT to use the parameters as specified by OP:
From the Gamma distribution wiki page we have that mean is $\alpha\beta$, standard deviation is $\beta\sqrt\alpha$ and the mode is $(\alpha-1)\beta$
So divide $\alpha\beta=10$ by $\beta\sqrt\alpha=5$ to get $\sqrt\alpha=2$, so $\alpha=4$ and $\beta=\frac{5}{2}$
The mode is $(\alpha-1)\beta=\frac{15}{2}$
The median cannot be calculated in a simple closed form, so you have to use some numeric method or software which can calculate the inverse of the lower incomplete gamma function $\gamma$.
$$F(x;\alpha,\beta)=\frac{\gamma(\alpha,x/\beta)}{\Gamma(\alpha)}=\frac{1}{2}$$
From where:
$$x=\beta\gamma^{-1}\left(\alpha,\frac{\Gamma(\alpha)}{2}\right)$$
OLD POST:
From the Gamma distribution wiki page we have that mean is $\frac{\alpha}{\beta}$, standard deviation is $\frac{\sqrt\alpha}{\beta}$ and the mode is $\frac{\alpha-1}{\beta}$
So divide $\frac{\alpha}{\beta}=10$ by $\frac{\sqrt\alpha}{\beta}=5$ to get $\sqrt\alpha=2$, so $\alpha=4$ and $\beta=\frac{2}{5}$
The mode is $\frac{\alpha-1}{\beta}=\frac{15}{2}$
The median cannot be calculated in a simple closed form, so you have to use some numeric method or software which can calculate the inverse of the lower incomplete gamma function $\gamma$.
$$F(x;\alpha,\beta)=\frac{\gamma(\alpha,\beta x)}{\Gamma(\alpha)}=\frac{1}{2}$$
From where:
$$x=\frac{1}{\beta}\gamma^{-1}\left(\alpha,\frac{\Gamma(\alpha)}{2}\right)$$
