Maximum Likelihood Estimator Question

[Pyroadept]

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!

 

[mighty2000]

Hi there,

Your approach seems correct to me. Since you are looking for the MLE of λ, you don't need to include the x_i's in the picture as they are just observed values and not parameters. Your final result of λ = 6/x also makes sense, as it represents the maximum likelihood estimate for the scale parameter given that the shape parameter is known to be 3. Good job!