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## Homework Statement

Let X

_{1}, X

_{2},...,X

_{n}be a random sample from the exponential distribution with mean θ and [itex]\overline{X}[/itex] = [itex]\sum^{n}_{i = 1}X_i[/itex]

Show that [itex]\overline{X}[/itex] ~ Gamma(n, [itex]\frac{n}{θ}[/itex])

## Homework Equations

θ = [itex]\frac{1}{λ}[/itex]

MGF Exponential Distribution = [itex]\frac{λ}{λ - t}[/itex]

MGF Gamma Distribution = ([itex]\frac{β}{β - t}[/itex])

^{α}

## The Attempt at a Solution

I've tried using the generating function of the exponential distribution but I end up with

[itex]\frac{(\frac{λ}{λ-t})^{n}}{n}[/itex]

I don't know what to do with the n in the denominator to get λ = [itex]\frac{n}{θ}[/itex]