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## Main Question or Discussion Point

In a certain country, the distribution of incomes in thousands of dollars is described by a gamma distribution with [itex]\alpha[/itex] = 2 and [itex]\beta[/itex] = 8. What is the probability that a man chosen at random will have the following incomes?

a. More than $14,000

b. At least $12,000

So I know that f(x;[itex]\alpha[/itex],[itex]\beta[/itex]) = [itex]\frac{1}{\Gamma(\alpha)\beta^{\alpha}}[/itex]x[itex]^{\alpha-1}[/itex]e[itex]^{-x/\beta}[/itex] for x>0,[itex]\alpha[/itex],[itex]\beta[/itex]>0

and [itex]\Gamma[/itex]([itex]\alpha[/itex])=[itex]\int[/itex][itex]^{\infty}_{0}[/itex]xe[itex]^{-x}[/itex] = 1

so f(x) = [itex]\frac{1}{64}[/itex] xe[itex]^{-x/8}[/itex]

Not sure where to go from here though.. Hopefully i'm heading in the right direction..

a. More than $14,000

b. At least $12,000

So I know that f(x;[itex]\alpha[/itex],[itex]\beta[/itex]) = [itex]\frac{1}{\Gamma(\alpha)\beta^{\alpha}}[/itex]x[itex]^{\alpha-1}[/itex]e[itex]^{-x/\beta}[/itex] for x>0,[itex]\alpha[/itex],[itex]\beta[/itex]>0

and [itex]\Gamma[/itex]([itex]\alpha[/itex])=[itex]\int[/itex][itex]^{\infty}_{0}[/itex]xe[itex]^{-x}[/itex] = 1

so f(x) = [itex]\frac{1}{64}[/itex] xe[itex]^{-x/8}[/itex]

Not sure where to go from here though.. Hopefully i'm heading in the right direction..

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