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Pyroadept
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Homework Statement
Lifetimes of components are Gamma distributed. The parameters of the Gamma are
shape = a
scale = λ
The pdf is:
f(x) = (λ^a).x^(a-1).e^(-λx)/Γ(a)
In this case, it is known that a = 3. Obtain the MLE of λ.
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far. To me it makes sense but if you could please take a look at it and see if I've gone wrong somewhere? It just seems a bit too straightforward...
--
L(λ;x) = f(x) ... no product as there are no x_i's, only one parameter, x
Sub in a=3 to eqn
L = (1/2).(λ^3).(x^2).(e^(-λx))/2 ... as Γ(3) = (3-1)! = 2! = 2
l = ln L = ln((x^2)/2) + 3.ln(λ^2) - λx ... where l is a lower-case L
∂l/∂λ = 0 + 6/λ - x
= 6/λ - x
Let ∂l/∂λ = 0 to find maximum
6/λ - x = 0
λ = 6/x
Can I leave it at that? Was I right to omit the x_i's from the picture? Thanks in advance for any help!