How to determine maximum likelihood estimator for this question

Question

confused as to go about this question,

it is asking to determine the MLE of this distrubtion

f(x,θ ) N=3,

X1..X2..X3

(x^3) . (e^-x/θ) / 6θ^4

where ^ = to the power of

Answer 1

To find the maximum likelihood estimator, we need to maximize the likelihood function, which usually consists of three things:

1) find the likelihood function

2) find the derivative of the likelihood function with respect to any parameters (in this case, θ)

3) set the derivative equal to zero

With the occasional intermediary step of taking the natural log of the likelihood function after finding it, since the natural log is a monotonic transformation, and the log of the likelihood function will have the same maximum as the original likelihood function.

Assuming

f(x,θ)=(x³) . (e^-x/θ) / 6θ⁴

Our likelihood function is f(x₁,θ)*f(x₂,θ)*f(x₃,θ) since we have three observations

We next take the log of this to get:

ln[(x₁³) . (e^-x1/θ) / 6θ⁴]+ln[(x₂³) . (e^-x2/θ) / 6θ⁴]+ln[(x₃³) . (e^-x3/θ) / 6θ⁴]

=3ln(x1)+-x1/θ-ln(6)-4ln(θ)+3ln(x2)+-x2/θ-ln(6)-4ln(θ)+3ln(x3)+-x3/θ-ln(6)-4ln(θ)

We take the derivative of this with respect to θ, and get

df/dθ=x1*θ^-2-4/θ+x2*θ^-2-4/θ+x3*θ^-2-4/θ

And set this equal to zero and solve for θ

0=x1*θ^-2-4/θ+x2*θ^-2-4/θ+x3*θ^-2-4/θ

Our final answer is

θ=1/12*(x₁+x₂+x₃)

Hope this helps!