please help me solve the following..
- Using the gamma distribution with E[X] fixed, show that Var[X] decreases as the shape parameter α increase.
- Given E[X] =1000 and Var[X] =〖500〗^2, find the appropriate parameter values for the Gamma and lognormal distribution.
There are various ways of describing the parameters of the gamma, so you may have to modify the notation below.
If the gamma is described using the shape parameter $\alpha$ and the rate parameter $\beta$, then we have $$E(X)=\frac{\alpha}{\beta}\quad\text{and}\quad \text{Var}(X)=\frac{\alpha}{\beta^2}.\tag{1} $$ These are standard formulas that I assume you are not expected to prove.
Suppose that $E(X)$ is fixed. Then $\beta=\frac{\alpha}{E(X)}$ and therefore $$\text{Var}(X)=\frac{(E(X))^2}{\alpha}.\tag{2}$$ It is immediate from (2) that as $\alpha$ increases the variance decreases.
The above formulas can be used to calculate the parameters of the gamma when expectation and variance are given. If your version of the gamma uses a scale parameter, use the fact that the scale parameter is the reciprocal of the rate parameter.